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This is to certify that the
dissertation entitled

POLYCRYSTALLINE DIAMOND THIN-FILM FABRY-PEROT
OPTICAL RESONATORS ON SILICON

presented by

Roger Allen Booth, Jr.

has been accepted towards fulfillment
of the requirements for the

Ph.D. Electrical Engineering

 

 

Major Professor’s Signature

M4 1.1% 2003

Date

MSU is an Affirmative Action/Equal Opportunity Institution

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LIBRARY

Michigan State

University

 

 

PLACE IN RETURN BOX to remove this checkout from your record.
TO AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

DATE DUE

DATE DUE

DATE DUE

 

gt)! 1 6 2008

 

092209

 

 

 

 

 

 

 

 

 

 

 

 

6/01 cJClRC/DateDue.p65-p.15

 

 

POLYCRYSTALLINE DIAMOND THIN-FILM FABRY-PEROT OPTICAL
RESONATORS ON SILICON

By

Roger Allen Booth, Jr.

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Electrical and Computer Engineering

2003

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ABSTRACT

POLYCRYSTALLINE DIAMOND THIN-FILM FABRY-PEROT OPTICAL
RESONATORS ON SILICON

By

Roger Allen Booth, Jr.

Micro-optical devices have applications ranging from gas detection to optical computing.
Michigan State University has the capability to deposit high optical quality diamond thin
films on silicon substrates using microwave cavity plasma reactors. Diamond is of
interest for optical applications due to its high index of refraction and wide spectral
transmission range. Combining polycrystalline diamond thin film deposition techniques
and MEMS fabrication techniques, an array of optical wavelength Fabry-Perot resonators

on a silicon wafer has been constructed and accurately modeled.

This thesis describes the first fabrication of diamond thin-film Fabry-Perot resonators on
silicon wafers. Standard fabrication techniques are employed, including thermal
oxidation, photolithography, PECVD diamond deposition, anisotropic wet etching, and
sputtering. The optical performance of the resonators is investigated by measuring the
through—transmission of the device as a function of wavelength. Performance is

correlated with the physical properties of the device, including surface roughness.

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Transmission of the device is simulated with multi—layer optics theory. A novel method
of incorporating surface roughness into standard multi-layer optics theory is developed in
the thesis, and applied to the resonators considering the surface roughness of the diamond
film to be modeled by Gaussian and non-Gaussian distributions. The models provide

valuable guidance into the design of the resonators.

Atomic Force Microscopy is employed to characterize the surface roughness of the films.
The films are found to be approximately Gaussian, but with non-negligible amounts of
skewness and kurtosis measured. A Pearson Type-IV distribution is used in the optical
simulation to represent non-Gaussian roughness in the diamond film. Use of this

distribution improves the fit between theory and experimental measurements.

iv

To my parents and my wife.

ACKNOWLEDGEMENTS

I would like to acknowledge my committee members: Dr. Reinhard, Dr. Hogan, Dr.
Grotjohn, Dr. Golding and Dr. Birge. Special thanks is due to Dr. Reinhard for guiding

me through this research and being a patient and excellent teacher.

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TABLE OF CONTENTS

List of Tables ..................................................................................................................... xi
List of Figures ................................................................................................................... xii
Chapter 1: Introduction ........................................................................................................ l
1.1 Introduction .................................................................................................................... 1
1.2 Objectives ...................................................................................................................... 2
1.3 Preview of the Thesis ..................................................................................................... 2
Chapter 2: Background Fundamentals ................................................................................. S
2.1 Introduction .................................................................................................................... 5
2.2 Fabry-Perot Resonators .................................................................................................. 5
2.3 Optical Properties of Diamond .................................................................................... 11
2.4 Modeling of Multi-Layer Optical Structures ............................................................... 15
2.4.1 MacLeod method ...................................................................................................... 16
2.4.2 Transfer Matrix Method ............................................................................................ 31
2.5 Effect of Interface Roughness on Transmission Through a Single Surface ................ 35

vi

3.3 C

3.3.1

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Chapter 3: Experimental Methods ..................................................................................... 46

3.1 Introduction .................................................................................................................. 46
3.2 Device Fabrication ....................................................................................................... 46
3.2.1 Fabrication Overview ................................................................................................ 47
3.2.2 Oxidation ................................................................................................................... 48
3.2.3 Photolithography ....................................................................................................... 51
3.2.4 Oxide Etch ................................................................................................................ 52
3.2.5 Diamond Deposition ................................................................................................. 54
3.2.6 Silicon Through-Etch ................................................................................................ 60
3.2.7 Gold Sputtering ......................................................................................................... 64
3.3 Optical Measurements ................................................................................................. 66
3.3.1 Measurement Procedure ............................................................................................ 66
3.3.2 Measurement Apparatus ........................................................................................... 67
3.4 Atomic Force Microscopy (AFM) .............................................................................. 70
3.5 Scanning Electron Microscopy (SEM) ....................................................................... 70
3.6 Summary ...................................................................................................................... 71
Chapter 4: Modeling Approach ......................................................................................... 73
4.1 Introduction .................................................................................................................. 73

vii

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4.2 Motivation for DeveIOping a Different Model ............................................................. 73

4.3 Model Developed for this Research ............................................................................. 76
4.4 Extension to Multiple Layers ....................................................................................... 80
4.5 Applying the Model ..................................................................................................... 83
4.5.1 Setting Up the Variables ........................................................................................... 83
4.5.2 The Gaussian Distribution ........................................................................................ 84
4.5.3 Calculating the Transmission .................................................................................... 89
4.6 Summary ...................................................................................................................... 93
Chapter 5: Experimental Results and Comparison with Theory Part 1 ............................. 95
5.1 Introduction .................................................................................................................. 95
5.2 Samples ........................................................................................................................ 95
5.3 Optical Constants ......................................................................................................... 98
5.4 Measurements on Diamond Membranes without Gold Coatings .............................. 101
5.5 Measurements on Fabry-Perot Cavities ..................................................................... 110
5.6 Using the Simulation to Gain Insight into the Device ............................................... 113
5.7 Summary .................................................................................................................... 120
Chapter 6: Experimental Results and Comparison with Theory 2 .................................. 122

viii

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6.1 Introduction ................................................................................................................ 122

6.2 Samples ...................................................................................................................... 122
6.3 AFM Images .............................................................................................................. 123
6.4 AFM Data .................................................................................................................. 127
6.5 Calculations of Statistics from AFM Image Files ...................................................... 129
6.6 Measurements ............................................................................................................ 131
6.6.1 AFM Image 1: RB2K8-l ........................................................................................ 132
6.6.2 AFM Image 2: RBZK8-2 ........................................................................................ 134
6.6.3 AFM Image 3: RB2K8-3 ........................................................................................ 135
6.6.4 AFM Image 4: FBZ6-1 ............................................................................................ 137
6.6.5 AFM Image 5: FB26-2 ............................................................................................ 139
6.6.6 AFM Image 6: FB26-3 ............................................................................................ 141
6.6.7 AFM Image 7: FB26-4 ............................................................................................ 143
6.6.8 AFM Image 8: FB26-5 ............................................................................................ 145
6.6.9 AFM Image 9: FB26-6 ............................................................................................ 146
6.6.10 AFM Image 10: RB2K9-l .................................................................................... 147
6.6.11 AFM Image 11: RB2K9-2 .................................................................................... 149
6.6.12 AFM Image 12: RB2K9-3 .................................................................................... 151
6.6.13 AFM Image 13: RB2K9-4 .................................................................................... 153

ix

6.6.14 AFM Image 14: RB2K9-5 .................................................................................... 155

6.6.15 AFM Image 15: RB2K9-6 .................................................................................... 157
6.6.16 Measurement Summary ........................................................................................ 159
6.7 Including Higher Order Moments in Optical Simulations ......................................... 163
6.8 Conclusions ................................................................................................................ 169
Chapter 7: Conclusions and Future Work ........................................................................ 173
7.1 Conclusions ................................................................................................................ 173
7.2 Future Work ............................................................................................................... 174
7.2.1 Future Fabrication Work ......................................................................................... 174
7.2.2 Future Work with Modeling ................................................................................... 176
Appendix A: MATLAB Programs ................................................................................... 180
Appendix B: Sample Data ............................................................................................... 185
Appendix C: Additional Measurements ........................................................................... 189

Tab

Tab

Tab

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LIST OF TABLES

Table 6.1 Summary of RB2K8 AFM measurements ....................................................... 159
Table 6.2 Summary of FB26 AFM measurements .......................................................... 160
Table 6.3 Summary of RB2K9 AFM measurements ....................................................... 161
Table B.3 List of samples created for this research. ................................................ 185-187
Table B.4 List of FB samples. ................................................................................. 187-188

xi

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LIST OF FIGURES

Figure 2.1 Diagram of an ideal Fabry-Perot resonator ........................................................ 6

Figure 2.2 Transmission of an ideal Fabry-Perot resonator with n=l, dzlum, and
R1=R2=0.9 ............................................................................................................................ 8

Figure 2.3 Optical transmission of Single Crystalline Diamond and CVD optical diamond

[ref. 24] .............................................................................................................................. 12
Figure 2.4 A wave encountering a material interface ........................................................ 21
Figure 2.5 a thin layer ........................................................................................................ 22
Figure 2.6 A multilayer stack of materials ......................................................................... 30

Figure 2.7 Theoretical calculation of transmission versus wavelength for a gold-diamond-

gold optical resonator with d=1um of diamond, and 25nm gold layers on each side. ...... 31
Figure 2.8 A single thin layer ............................................................................................ 32
Figure 2.9 The general scattering geometry ...................................................................... 36
Figure 2.10 Cross-section of a rough interface .................................................................. 38
Figure 2.11 Cross-section of a rough interface .................................................................. 40

Figure 3.1 Cross-sectional view of two Fabry-Perot resonators. Drawing not to scale....47

Figure 3.2 Cross-sectional view of the wafer after oxidation. Drawing not to scale. ....... 51

xii

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Figure 3.3 Cross section of the wafer after oxide etch and photoresist removal. Drawing

not to scale. ........................................................................................................................ 54
Figure 3.4 Patterned oxide on 2-inch diameter wafer ........................................................ 55
Figure 3.5 Cross section of wafer after diamond deposition. Drawing not to scale. ........ 57

Figure 3.6 Wrinkled window resulting from low deposition temperature. This is an
optical image taken with dark-field illumination. .............................................................. 58

Figure 3.7 Flat window shown for comparison to Figure 3.6. This is an optical image

taken with dark-field illumination. .................................................................................... 58
Figure 3.8 Cross section of the wafer after the silicon etch. Drawing not to scale ........... 60
Figure 3.9 Schematic of the apparatus used for KOH etching of silicon. ......................... 61

Figure 3.10 Lookin g through a clear diamond window onto a transistor (transistor shown
for purposes of illustration only) ....................................................................................... 63

Figure 3.11 Holes seen in wafer are actually flat, transparent diamond windows. ........... 64

Figure 3.12 Diagram of the optical measurement apparatus constructed for this

research. ............................................................................................................................. 69
Figure 4.1 a smooth slab .................................................................................................... 74
Figure 4.2 Two paths through a rough thin film ................................................................ 76

Figure 4.3 Two transmission plots for a rough diamond slab. In both cases, thickness =
lum, roughness = 20nm. The difference between using the approach of equation [4.6]
and that of using [4.12] is readily apparent ........................................................................ 80

Figure 4.4 A multilayer structure with a rough interface ................................................... 81

xiii

 

Fig

 

Figure 4.1 Simulation error as a function of dpp .............................................................. 87

Figure 5.1 Deposition Temperature as a function of Deposition Pressure for MCPR
Configuration 7 with a 59mm quartz tube height and approximately lkW of incident

power .................................................................................................................................. 97
Figure 5.2 Data for the Index of Refraction for gold as a function of wavelength. ......... 100
Figure 5.3 Data for the Absorption Constant of gold as a function of wavelength ......... 101

Figure 5.4 Transmission versus wavelength for diamond window 1 on wafer RB2K9.
The wavelength axis spans from the UV (200nm) into the IR (2500nm). ...................... 102

Figure 5.5 Transmission for the same window as shown in Figure 5.3, diamond window
1, but taken with Bausch & Lomb equipment. ................................................................ 104

Figure 5.6 Transmission versus wavelength for diamond window 2, on wafer
RB2K8. ............................................................................................................................ 105

Figure 5.7 Transmission versus wavelength for diamond window 3, on wafer
RB2K8. ............................................................................................................................ 106

Figure 5.8 Transmission versus wavelength for diamond window 4, on wafer
RB2K9. ............................................................................................................................ 107

Figure 5.9 Transmission versus wavelength for diamond window 5, on wafer
RB2K9. ............................................................................................................................ 108

Figure 5.10 Transmission versus wavelength for diamond window 6, on wafer
RB2K9. ............................................................................................................................ 109

Figure 5.11 Transmission through Fabry-Perot resonator l, constructed on wafer
RB2K9. ............................................................................................................................ 111

Figure 5.12 Transmission through Fabry-Perot resonator 2, on wafer RB2K9. .............. 112

xiv

 

 

 

 

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Figure 5.13 Q as a function of surface roughness for a resonator based on a 1pm thick
diamond film. ................................................................................................................... 114

Figure 5.14 QT as a function of gold thickness for a resonator based on a 111m thick
diamond film. ................................................................................................................... 115

Figure 5.15 QT as a function of gold thickness for a resonator based on a lum thick
diamond film with a surface roughness of 18nm RMS. .................................................. 116

Figure 5.16 Q as a function of surface roughness for a resonator based on a 111m thick
diamond film. ................................................................................................................... 117

Figure 5.17 Q as a function of surface roughness for both gold layer thicknesses
considered. ....................................................................................................................... 1 18

Figure 5.18 Peak transmission vs. surface roughness for the two gold layer thicknesses
considered. ....................................................................................................................... 1 19

Figure 6.1 AFM picture of sample RBZKS (this image will also be referred to as AFM
Image 2) ........................................................................................................................... 124

Figure 6.2 AFM image of the sample FB26 (this image will also be referred to as AFM
Image 8). .......................................................................................................................... 125

Figure 6.3 AFM image of sample FB26 (this image will also be referred to as AFM
Image 9). .......................................................................................................................... 126

Figure 6.4 Simulated 3-Dimensional view of data in Figure 6.1, sample RB2K8. All axes
are in units of micrometers. (afmthing2.m, then use “mesh(x1,y4,c)” and scale
appropriately) ................................................................................................................... 127

Figure 6.5 Distribution of surface heights for Figure 6.1 (data points) and an ideal
Gaussian distribution (black lines) ................................................................................... 129

Figure 6.6 AFM Image 1 ................................................................................................. 132

XV

Figure 6.7 Distribution for AFM Image 1 ....................................................................... 133

Figure 6.8 Distribution for AFM Image 2. ...................................................................... 134
Figure 6.9 AFM Image 3. ................................................................................................ 135
Figure 6.10 Distribution for AFM Image 3. ................................................................... 136
Figure 6.11 AFM Image 4. .............................................................................................. 137
Figure 6.12 Distribution for AFM Image 4. .................................................................... 138
Figure 6.13 AFM Image 5. .............................................................................................. 139
Figure 6.14 Distribution for AFM Image 5. .................................................................... 140
Figure 6.15 AFM Image 6. .............................................................................................. 141
Figure 6.16 Distribution for AFM Image 6. .................................................................... 142
Figure 6.17 AFM Image 7. .............................................................................................. 143
Figure 6.18 Distribution for AFM Image 7. .................................................................... 144
Figure 6.19 Distribution for AFM Image 8. .................................................................... 145
Figure 6.20 Distribution for AFM Image 9. .................................................................... 146
Figure 6.21 AFM Image 10. ............................................................................................ 147
Figure 6.22 Distribution for AFM Image 10 ................................................................... 148
Figure 6.23 AFM Image 11. ............................................................................................ 149

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Figure 6.24 Distribution for AFM Image 11 ................................................................... 150

Figure 6.25 AFM Image 12. ............................................................................................ 151
Figure 6.26 Distribution for AFM Image 12 ................................................................... 152
Figure 6.27 AFM Image 13 ............................................................................................. 153
Figure 6.28 Distribution for AFM Image 13 ................................................................... 154
Figure 6.29 AFM Image 14 ............................................................................................. 155
Figure 6.30 Distribution for AFM Image 14 ................................................................... 156
Figure 6.31 AFM Image 15 ............................................................................................. 157
Figure 6.32 Distribution for AFM Image 15 ................................................................... 158

Figure 6.33 Plot of the constraint on Kurtosis as a function of Skewness for Equation
[6.6]. ................................................................................................................................. 166

Figure 2.34 Measured distribution from AFM Image 11 compared to calculated Pearson
Type-IV distribution. ....................................................................................................... 167

Figure 6.35 Measured and simulated transmission through a diamond window, simulation
includes surface roughness, skewness, and kurtosis. ....................................................... 168

Figure 6.36 Measurement and simulation of a Fabry-Perot resonator, with surface
roughness, skewness, and kurtosis considered. ............................................................... 169

Figure 6.37 Comparison of simulations using a pure Gaussian, and a Pearson Type—IV
distribution. ...................................................................................................................... 170

xvii

Figure C.3 Diamond window on the RB2K9 wafer measured with Perkin-Elmer UV-Vis
system. 0.66pm thickness. .............................................................................................. 189

Figure C.4 Diamond window on RB2K9 measured with Perkin—Elmer UV-Vis system.
0.695pm thickness. .......................................................................................................... 190

Figure C.5 Diamond window on RB2K8 measured with Bausch & Lomb system.
Approximate thickness 1.55pm. ...................................................................................... 191

Figure C.6 Diamond window on RB2K8 measured with Bausch & Lomb system.
Approximate thickness 1.62um. ...................................................................................... 192

Figure C.7 FP Cavity on RB2K8 as measured with Perkin-Elmer UV-Vis system.
Diamond film is 1.59pm thick, with 12nm and 20nm gold coatings. ............................. 193

Figure C.8 FP Cavity on RB2K8 as measured with Perkin-Elmer UV-Vis system.
Diamond film is 1.55pm thick, with 12nm and 25nm gold coatings. ............................. 194

Figure C.9 Diamond window on RB2K8 measured with Bausch & Lomb system.
Approximate thickness 1.55pm. ...................................................................................... 195

Figure C.10 FP Cavity on RB2K8 measured with Bausch & Lomb system. Approximate
diamond thickness 1.75mm. Gold coatings approximately 12nm and 25nm thick. ....... 196

Figure C.11 FP Cavity on RB2K8 measured with Bausch & Lomb system. Approximate
diamond thickness 1.57mm. Gold coatings approximately Onm and 15nm thick. ......... 197

xviii

 

 

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Chapter 1: Introduction

1.1 Introduction and Motivation

This thesis is written on the subject of a polycrystalline diamond optical device,
integrated onto a silicon wafer using nricrofabrication techniques. It describes and
reports on the fabrication of diamond Fabry-Perot resonators on silicon, and uses
multilayer optics theory to model the optical performance of the device with respect to

the physical parameters of the device.

The general area of on-chip optics is of interest for a variety of applications. From
simple optical detectors to full-scale optical interconnects for devices, on-chip optics is a
broad area prorrrising many applications and continued research interest. Because of the
semiconductor process engineer’s general affinity for applying thin films to silicon and
other semiconductor substrates, the area of multi-layer thin-film optics seems almost a
natural partner for on-chip optics. Indeed, this is the case for the devices constructed in

this research.

Diamond is of particular interest for optical applications because of its uniquely broad
window of transmission in the electromagnetic spectrum. Moreover, except for high
temperatures in oxygen rich atmospheres, it may be used in a variety of hostile

environments. A contribution of this research is the integration of diamond optical

 

 

 

 

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devices, specifically Fabry-Perot resonators, onto a silicon wafer. This is accomplished

by combining MEMS fabrication technology with diamond deposition techniques.

1.2 Objectives

The first objective of this work is to establish a repeatable fabrication procedure for the
optical resonators. Secondly, a method of optically characterizing the samples shall be

established.

Next, a mathematical model of the optical performance of the device must be
constructed. This model is to be based on physically measureable properties of the
device. Finally, the mathematical simulation should be capable of guiding the future

direction of this work.

1.3 Preview of the Thesis

Chapter 2 presents a review of background material relevant to two main parts of this
thesis, the optical properties of the diamond film used in the device, and the modeling of
the optical performance of the device. The optical properties of diamond are discussed
briefly, mainly relating to the polycrystalline diamond films used in this research. The
modeling material includes sections on ideal Fabry-Perot resonators, two sections on
multi-layer optics theory, and a discussion of surface roughness and its impact on a single

interface with respect to optical reflection and transmission at that interface.

 

 

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Chapter 3 presents a detailed explanation of the experimental procedures used for this
work, including the device fabrication and characterization. The fabrication of the device
uses some MEMS style fabrication sequences to create thin, free-standing diamond
windows on the silicon wafer. These windows form the basis of the Fabry-Perot
resonators investigated in this research. The device is characterized optically by passing
light through the diamond windows and observing the transmission characteristics of the
film. Chapter 3 discusses how these measurements are performed. Additional
characterization techniques are employed, namely SEM and AFM to evaluate the

properties of the diamond film depositions. These techniques are also briefly introduced.

Chapter 4 presents the model developed for this work, and explains a numerical
implementation of the model used extensively in this thesis. The model combines the
multi-layer optics theory and the surface roughness modeling presented in Chapter 2.

This allows improved modeling of the optical transmission of the device.

Chapter 5 compares optical transmission measurements to simulations based on the
model presented in Chapter 4. The measurements and simulations in Chapter 5 show
good correspondence, but lead to some conclusions not expected at the outset of this

research.

Chapter 6 takes a detailed look at surface roughness of the diamond films grown for this

work, and investigates their deviation from the Gaussian distribution used in Chapters 4

and 5, and discusses how this deviation impacts the simulation. It is found that the

inclusion of a more sophisticated distribution can improve the performance of the model.

Chapter 7 outlines the conclusions drawn in the preceding chapters, and discusses some

possible future directions for this work.

 

 

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Chapter 2: Background Fundamentals

2.1 Introduction

This chapter presents background material that is essential to various aspects of the
research performed in this dissertation. First, the operating principles of Fabry-Perot
resonators are described, as well as several performance measures of resonators.
Examples of previous investigations of on-chip Fabry-Perot resonators are described.
Next, the optical properties of diamond are reviewed, particularly in the context of thin
film polycrystalline diamond formed by chemical vapor deposition (CVD). The optical
structures in this research are analyzed by a matrix method, and the background
mathematical formulation of this method is described, beginning with Maxwell’s
equations. Finally, the effects of a rough surface on reflection and transmission are
reviewed, primarily in the context of a superposition of plane waves and for normal

incidence.

2.2 Fabry-Perot Resonators

The Fabry-Perot resonant cavity is an important device for a variety of optical and
microwave applications, including highly selective band—pass filters. As shown in Figure
2.1, an ideal optical Fabry-Perot cavity consists of a non-absorbing optical medium with

partially reflective, non-absorbing mirrors on either side with perfectly smooth interfaces

 

 

 

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between the optical medium and the mirrors. The ideal cavity will allow up to 100%

transmission of certain wavelengths of light, while heavily attenuating other wavelengths.

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Figure 2.1 Diagram of an ideal Fabry-Perot resonator

 

 

Elaboration on the following review of the Fabry-Perot resonator can be found for
example in Verdeyor‘. Resonance of the cavity occurs when there is an integral number
of half-wavelengths between the two mirrors. This resonance corresponds to the
wavelength of peak transrrrission. The transmission of a Fabry-Perot resonator can be

calculated as a function of the freespace wavelength, 20, to be:

T- (1-R1)-(1-R2) [,1]

(1—./R1R2)2 + 4./R1R2 sin2[2;:in]

where n is the refractive index of the optical medium, R1 and R2 are the power

 

reflectivities of the lossless rrrirrors and dis the distance between the rrrirrors. The
derivation of an equation equivalent to [2.1] is shown in Chapter 4. The maximum

transmission as a function of the mirror reflectivities is given by:

 

 

 

 

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Therefore when R, and R2 are equal, the cavity will transmit 100% of the incident power

[2.2]

at the resonant wavelengths, 2,", where

d = m—’"— [2.3]

and m is an integer which identifies the resonant mode. In equation [2.3], the
wavelength, /l,,., refers to the wavelength in the optical medium, i.e. the free space
wavelength divided by n. Figure 2.2 shows the result of equation [2. 1] plotted for

R1=R2=0.9, d=1um and air (n=1) as the medium between the mirrors.

 

 

 

 

 

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Figure 2.2 Transmission of an ideal Fabry-Perot resonator with n=l, d=1um, and R1=R2=0.9

There are several useful figures of merit for Fabry-Perot resonators. One is the quality

factor, Q, which is related to the sharpness of the transmission peaks by:
Q = —m- [24]

where vmis the resonance frequency of the mth mode and Avm is the spectral width, or full

width at half maximum of the peak at vm.

Using equation [2.1], Q can be expressed in an analytical form as a function of cavity

parameters and resonance wavelength as:

 

 

 

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Q 4m 1 [2-5]
1“(R1R2)2
The photon lifetime inside the cavity, Ip, is the time for a round trip divided by the
fraction of photons lost to the cavity per round trip. It is related to Q by:
z”, = -——Q—- [2.6]
mm

where (am is the angular frequency of the peak of mm mode. IP for an ideal cavity can be

expressed in terms of cavity parameters by:

2 d/
7p = __n___c_ [2.7]
1 — R1R2
where the numerator of this expression is the time it takes a photon to make a round trip

in the cavity, and the denominator is the fraction of photons lost per round trip. Other

loss mechanisms in a non—ideal cavity would lead to an expression of the form:

 

2nd /
IP = c [2.8]
1— Rle + (other losses)
Another measure of resonator performance is the separation between neighboring
resonant peaks. It is known as the free spectral range and is expressed as:
c / n
V = — [2.9]
f 2d

Finally, the finesse is defined as the ratio of the free spectral range to the spectral width,

which can also be related to Q:

l
V _
= f =”(R1R2)4 ___ ’10 Q [2.10]
AV," 1— RIRZ 2nd

For the ideal cavity illustrated in Figure 2, the free spectral range is equal to l.5*10l4 Hz
and the spectral width of the m=2 mode is approximately 5* 10'2 Hz. By equation
[2.10], this corresponds to a finesse of 30 for this peak. Using equation [2.5], Q is

approximately 60 for the same peak.

As discussed at the beginning of this section, Fabry-Perot resonators form wavelength-
selective band-pass filters. Such a filter in and of itself can be useful in certain
applications, such as detecting an optical signal broadcast at a certain wavelength. In this
case, the resonator would serve to filter out other wavelengths, so that only the
wavelength carrying the signal would reach the detector. Another interesting application
is an array of resonators on the same substrate, each tuned to a different wavelength, such
that an on-chip spectrometer could be formed. Such a device could be used, for example,
to investigate the chemical composition of a gas based on the optical wavelengths
absorbed and transmitted through the gas. Another application of an on-chip Fabry-Perot

resonator is a pressure transducerz.

Ahmadi3 et al have reported on integrating Fabry—Perot resonators onto semiconductor
substrates, using CMOS and MEMS technology. Booth4 et al have reported on the first
diamond film Fabry-Perot resonator using standard Plasma-Enhanced CVD and MEMS

processes.

Correias’fi‘7 et al, and Bartekg‘9 et al have reported on constructing an array of 16 Fabry-

Perot resonators and have produced working on-chip spectrometers. Their devices use

10

 

 

ff.

"1

If“)

)1

If

C).

 

SiOz as the medium between the mirrors. Partially transparent silver is used to form the
mirror coatings. The results achieved are good with reported measured finesse values of
12, although surface roughness of the oxide film can become an issue with this technique,
as simulations give finesse values of 40 for ideal structures. Additionally, if the distance
between the reflective surfaces of the Fabry-Perot resonator can be adjusted, the cavity

1 . . .
can be tuned. Several groups 0 H '2 '3

are using MEMS technology to achieve this.
In addition to using Fabry-Perot resonators to create a spectrometer, it is possible to use a
diffraction grating to separate an optical signal into its constituent wavelengths. This will

not be discussed here, but it will be noted that some groups are working on this”.

2.3 Optical Properties of Diamond

When constructing an optical resonator, absorption in the optical medium between
reflectors can affect resonator performance as implied by the previous section 2.1. For
example, $02, which absorbs heavily in portions of the infrared (IR), would not be a
good Fabry-Perot medium if the device were needed to operate over a wide span of IR
wavelengths. Likewise, silicon would not work well in the visible portion of the
spectrum, due to its small band gap and accompanying absorption of visible light.
Diamond is a particularly attractive material for broad—based Fabry-Perot applications,
because it is transparent from the ultra violet (UV) into the microwave portion of the

spectrum with very slight absorption in the IR.

11

 

 

 

 

 

$0
the
ab;

fol

EX:
“:1

itd

Diamond is considered to have the widest optical spectral transmission range of any
known solid material15 . Figure 2.3, from Harris, shows the optical transmission of single
crystalline diamond plotted with the transmission of high quality CVD diamond. This
spectral range can ideally span from around the band gap energy of 5.47eV to the m

portion of the electromagnetic spectrum.

 

Please see Ref. 24 (Harris)

 

 

 

Figure 2.1 Optical transmission of Single Crystalline Diamond and CVD optical diamond [ref. 24]

Some sub-bandgap absorption can be seen in high quality diamond. This is due in part to
the Urbach rule16 which models a temperature-dependent exponential increase in
absorption near the bandgap. Additional absorption is seen near the bandgap, which

follows a temperature-independent relationship to about 0.35 um”.
Excitation of vibrational modes in carbon-to-carbon bonds leads to weak absorption of

wavelengths in the mid-infrared”. Since pure diamond is a symmetric covalent material,

it does not posses a dipole moment, which would lead to single phonon absorption in the

12

IR regime. Multi-phonon processes cause weak absorption between 2.5 and 7.5um. The
largest multi-phonon absorption peak magnitude is about 12 cm", which occurs at about

Sum.

This research primarily studies the region of the spectrum between 700nm and 1600nm.
For this spectral range, high optical quality CVD diamond should show virtually no

optical absorption.

For this research, the index of refraction for diamond as a function of wavelength is

modeled by the commonly used Sellmeier equation for diamond"):

 

 

 

4.33502.2 03306-22
"d( )=\/ [2.11]

+
,12 - (0.1060)2 .12 — (0.1750)2
where A is in units of micrometers. This empirical equation is valid over the range of

wavelengths studied in this research, as well as much more of the electromagnetic

spectrum.

Like any crystalline material, imperfections in diamond’s crystal structure can influence
the optical pr0perties exhibited by a particular sample”. The most common defects
found in natural diamond are the presence of nitrogen, and to a lesser extent, boron”.

The presence of nitrogen typically leads to absorption of higher photon energies. Such
absorption typically leads to a yellow or brown appearance of the diamond. Boron,
which occurs less commonly than nitrogen in natural diamond, leads to IR absorption that

can extend into the longer wavelengths of the visible spectrum. Absorption at the longer

13

 

 

-r_-3

Cl.

\1

ill

['0‘

I10

0P

 

C)
but

H}-

visible wavelengths leads to a blue appearance. Also, if the incorporation of nitrogen or
boron into the crystal breaks the lattice symmetry, single phonon absorption in the

infrared may occur”.

The diamond used in this research is polycrystalline diamond deposited on a substrate by
plasma enhanced chemical vapor deposition (CVD). For modeling purposes, though, the
diamond film is treated as a continuous slab of diamond, i.e. grain boundary effects
within the diamond are not explicitly considered. For the films in this study, the
excellent match of empirical data to theoretical calculations can be argued to justify such

an assumption.

It should be noted, however, that not all diamond produced by CVD techniques is of high
optical quality. The CVD technique itself may introduce chemical impurities, surface
roughness, non—diamond bonds between carbon atoms, and grain boundaries that would
not be present in pure, ideal diamond. Each of these may cause characteristics in the

optical properties of the CVD diamond that would not be present in ideal diamond”.

CVD may lead to the presence of nitrogen or boron as can be found in natural diamond,
but an additional concern with CVD diamond is the presence of hydrogen and oxygen.
Hydrogen and oxygen can often be found in the feed gases used in the CVD process, and
as such, may be incorporated into the film during growth. Hydrogen in the diamond
lattice often allows additional phonon modes, centered around 3.50m. Oxygen is

sometimes added to CVD feed gasses to improve the properties of the diamond at visible

14

wavelengths by reducing the number of non-diamond carbon to carbon bonds. Oxygen

incorporated into the film, though, will lead to increased absorption in the IR”.

Carbon-to-carbon bonds in diamond are referred to as sp3 bonds. Depending on CVD
growth conditions, varying amounts of non-sp3 bonds can be introduced into the lattice.

This can lead to sub-bandgap absorption in the visible range22‘23.

Typically, producing films of high optical quality means that process parameters are such
that growth rate will be relatively low“. High growth rate conditions often lead to
various defects in the resultant diamond as described above. Additionally, as-grown
surface roughness of CVD films can vary greatly. Surface roughness can lead to great
reductions in the optical performance of a film25'26'27. However, polished slabs of CVD
polycrystalline diamond24 can demonstrate optical properties quite close to the best single

crystal diamond from the UV, through the visible, and into the IR and microwave.

2.4 Modeling of Multi-Layer Optical Structures

The theoretical calculations in this research require the treatment of multilayered optical
structures. For many years, the n-layer problem in optics was considered to be
intractable, much like the n-body problem in mechanics. In 1937, however, Rouard
showed a method for closed form matrix solution in which multilayer structures can be
analyzed by representing each layer by a 2 x 2 matrix28'29’30’3"32. This method has

become essential to the analysis of multilayer thin film optics. Subsequent treatments

15

 

Ill?

no.

Cit.

Th

0ftw

 

have resulted in different, but equivalent, formulations of this method. Two distinct
formulations have been used in the analysis of results in this research, and are described

in the following sections.

The first formulation shall be referred to here as the MacLeod technique, after the
reference most often cited in the literature when referring to this technique“. The second
technique, commonly called the transfer matrix, uses matrices containing the Fresnel
coefficients and phase propagation across the layer. This nomenclature is only partially
consistent with the literature, as MacLeod’s method is sometimes called a ‘transfer
matrix’ method as well. However, the two methods are distinct enough that inspection of
the equations presented should readily identify which technique is being used, regardless

of the particular nomenclature chosen by the author.

Each of the techniques presented relies upon a superposition of plane waves, and thus
linearity of the optical medium is assumed. Additionally, this research will also assume
normal incidence of the light onto the optical structure. Although the models can treat
non-normal incidence, the assumption of normal incidence simplifies the derivations and
calculations to some extent and is an accurate representation of the experimental set-up.

The reader is referred to the cited references for the case of non-normal incidence.

2.4.1 MacLeod method

The development of MacLeod’s approach begins with Maxwell’s equations, which are

often expressed in Gaussian units for work with thin film optics33'34. Gaussian units will

16

 

 

M

He

fo

qr:

Ch;

 

be briefly discussed here. The velocity with which light propagates in free space is taken

to be:

1

«“050

In the MKS system of units, 110 is chosen as 411;*10'7 Henries/meter. However, in

 

c = [2.12]

Gaussian units, 110 is chosen as unity. For optical materials, the total magnetic

permeability of the material is almost always essentially unity.

The Poynting vector in Gaussian units is written as:

C
S=—E H 2.13
47r( X ) [ ]

Maxwell’s equations written in these units are:

Vsz-lgg

cat

47:. 18D
VXH=—-—J+-— [2.14]
c c a:

V-B=0
V-D=47r,o

Here E is the electric field vector, B is the magnetic induction vector, H is the magnetic
vector, j is the electric current density, D is the electric displacement, c is the speed of
light, and p is the electric charge density. Assuming a source free region with no space

charge:

17

 

 

 

fr

([9.

VXE=—l§§

cat

VxH=flj+l§R [2.15]
c c a:

V-B=0
V0D=0

Additionally, a homogeneous, isotropic medium is assumed such that:

D=£E

B=aH [2.16]
i=0E

Maxwell’s equations are then expressed as:

V><E.-ޤE
c at
VxH=flE+£a—E [2.17]
C C t
V-B=O
V-D=O
Working with the first equation in [2.17]:
VxVxE=Vx(—fla—H)=—fi2—(VXH) [2.18]
c at c a:

substituting the second equation of [2.17] into [2.18] leads to an expression for the B

field:

8_,uazE + 47mg _BE

V2E=
c2 dtz c2 at

 

[2.19]

A plane wave solution to this equation, propagating in the z-direction with velocity v, is

given by:
imp—E]
E = Eoe v [2.20]

18

where 6015 the angular frequency. This solution requires that:

 

[2.21]

This requirement can be seen simply by inserting equation [2.20] into equation [2.19] and

carrying out the operations of equation [2.19]:

i t—Ej 82 MIG-E) 4 a iu{t—£)
VzEem( v =EE—Ee v + flay—Be v 2.22
0 c2 dt2 0 c2 at 0 l ]

 

After carrying out the operations and canceling common terms:

 

[2.23]

Equation [2.23] is the same as equation [2.21]. The ratio of c/v is the index of refraction.
MacLeod defines the optical (or characteristic) admittance, N, as the c/v ratio“, which, by

equation [2.21], leads to:

 

[2.24]

This means that N is a complex variable, which in general can be written as a sum of a
real and complex part:

N = n - ik [2.25]
Although n and k could be negative in principal, this would not correspond to a physical
situation, so It and k are assumed to be positive. MacLeod defines n as the refractive
index and k as the extinction coefficient, which is related to the optical absorption

coefficient, a, as:

a = -—— [2.26]

Next, equation [2.25] is inserted into equation [2.24].

47mm

N2 = (n—ik)2 = n2 —i2nk -k2 = sp-i [2.27]

The real and imaginary parts of equation [2.27] are equated, and the assumption that u is

unity for optical materials is invoked, giving the following two relations:

n2 — k2 = e [2.28]
4
2nk = E [2.29]
60
By noting that 0) can be expressed as:
2m:
0) = — [2.30]
x1

and using the definition of N as the ratio of c/v, the expression for the plane wave,

equation [2.20], can be rewritten as:

E = E0 exp[i(a)t — (05)] = E0 expl:i(wt — ifs/El] [2.31]
v

Using equation [2.25] for N in equation [2.31] shows how the extinction coefficient k

works:

E = E0 exp[i(ax _. Mj] = E0 CXP|:— 2%”(_ Z] CXp|:l[C€I — 33—02“ 2]] [2.32]

Since the exponential term containing k is real and negative, it causes the amplitude to

decay as the wave propagates in the z direction.

MacLeod next shows a relationship between E, H, and N which is very important to the

development of this matrix technique:

20

 

H = N [2.33]
rXE

where r is a vector perpendicular to E and H and for this simple derivation, will be

considered the direction of the wave’s propagation. MacLeod next introduces a modified
admittance, 11, although for the case of a normal incidence as assumed at the outset of this
section, the modified admittance is exactly the same as the optical admittance, N. So, for

purposes of this work, equation [2.33] is rewritten as:

H = N(er) [2.34]

Next, equation [2.34] is applied to a simple material boundary in order to see an
important relationship used in the matrix technique. Also, superposition is used to break

the total field into positive- and negative-going waves.
130+ Eo'

material 0

 

+ material 1
E1

Figure 2.4 A wave encountering a material interface

Figure 2.4 shows a wave, E0+, propagating in direction r, incident on a material interface,

marked by line a, between material 0 and material 1. E0' is the reflected wave, and EC is

21

the transmitted wave. Equation [2.34] can be used to relate the electric and magnetic
fields by:

H+ = N (r X EL)

0 0 0 [2.35]

H6 = NO(-rXE6)
MacLeod then goes on to show the derivation of the Fresnel reflection and transmission
coefficients, as well as how to handle the case of non-normal incidence. However,
equation [2.35] is sufficient to begin the derivation of the multilayer matrix approach.
The model will initially be derived for a single layer, and then extended to the case of

multiple layers. Figure 2.5 shows a wave incident upon a single layer.

 

 

Eo+ 130' r
material 0
a
E" _ _
E1 materral 1 d
b
E; X material 3
Figure 2.5 a thin layer

Using superposition, the fields in material 1 at boundary b can be written as:

22

111,, = 11;), +H;,, = Nllerfb)-N1(er,—b)

+ _ [2.36]
Ew=Ew+Ew
The electric field equation can be rewritten as:
er1b = erfb + erfb [2.37]

Equations [2.36] and [2.37] can be combined to eliminate the negative-going portion of

the wave, which gives:

Hlb = Nlierfb)— erl'XEib —erfb) [2.38]

which simplifies to:

2erfb = 9N1 +er1b [2.39]
1

Similarly, the positive-going portion of the wave can be eliminated to give:

ZYXEl—b = -%+YXElb [2.40]
1

Next, the fields in material 1 at interface a are written in terms of the fields at interface b
by using a propagation term. The propagation term is based on equation [2.31] and the
thickness of the layer, (1. If interface b is located at 2, then interface a is located at z-d

such that:

23

 

E(z — d) = E0 exp|:i[a)t — ”ME: _ ‘10)] [2.41]

Also, this propagation term gives rise to the concept of a “thin” layer, in that it is

assumed that the time dependence of the wave is not apparent over the thickness of the

film.

 

 

E(z - d) z E0 expl} i( 2723,12 - 27d:1d):| = exp[i 2711:1d ]E(z) [2.42]

MacLeod defines the phase factor as 5, so that:

 

E(z — d) = ei6E(z) [2.43]
with:
6 = 277:, ‘d [2.44]

With this in mind, the fields at interface a can now be written conveniently in terms of

the fields at interface b:

+ + £6
Ela = Elbe

_ _ _‘5

E1 = Elbe l

H+a H+ I6 [245]
la = lbe
__ _ _-§

Hla = Hlbe I

Using equations [2.39] and [2.40], the electric fields are then:

24

1a
1 [2.46]
_ 1 H _-
rXEla = — -——lb-+l'XElb e 16
2 N1
Equation [2.36] can be used to find expressions for the magnetic fields:
N H -
Hfa = Nllerf'a): ——1 l+rXElb e'5
2 N1
[2.47]

H]; = —Nl(er;a )= “921%-? ”XI-5311,)?”
l

The total electric field is next written from equation [2.46]:

_ 1 H - 1 H _-
er =er+ +er =— 4’4.er e'5+— ———1!’—+er e '5
la la la 2[ N1 1b] 2 N1 lb

[2.48]
1 H . _. . _.
= — —1—b(e'6 -e '6)+rXE1b(e'6 +6 '6)
2 N1
By the applying the Euler identity, this can be simplified further to give:
Hlb . .
erla =er1b(cos5)+—N—(tsrn5) [2.49]

1

Similarly, the total magnetic field can be found to be:

25

11161:"!Vi h+rXElb eié—fll‘ -§—&+rXElb e—i6

= 9311’— (eia + 846 )+ %r x Elb (em - 8‘46) [250]
= Nler1b(isin §)+ Hlb(cos 5)

The results of equations [2.49] and [2.50] can be combined in matrix from:

[erla]_ cosd i—sind [IXElb] [251]

H1“ iNl sin 5 605 6 Hlb
The 2 x 2 matrix in equation [2.51] is called the characteristic matrix for the film. This
matrix is a function of the properties of the film, namely, the index of refraction, N, and
the film thickness, through equation [2.44]. MacLeod then invokes equation [2.33] to
eliminate the magnetic field term, while introducing the admittance of the assembly, Y,
defined as the ratio of the magnetic field to the wave vector crossed with the electric field
all at boundary a, and using the fact that the electric field parallel to the material

boundary is continuous across the boundary:

1 _i. ' 1
rXanirj = C055 N1 SW5 [N ]er2,, [2.52]
ileind c056 2

Two new variables, B and C, are defined as the components of the product:

B _i— ° 1
[ ]: c036 N1 srnd [ ] [2.53]
C ileincS' 0035 N2

26

Equation [2.53] has all of the information needed to calculate transmission through,
reflection from, and absorption in the layer. First, the reflection from the layer will be

studied. The relationship between Y, C and B is:

Y = — [2.54]

Since Y is an equivalent admittance, it can be used in the calculation of the Fresnel

reflection coefficient, r, as:

 

[2.55]

The power reflection, R, is then:

R=r-r*= N—O:—Y- - [VD—Y [2.56]
N0+Y N0+Y

In calculating the transmission, first the mean value of the Poynting vector is needed,

 

which is:

S = [é Re(EH*)] . r [2.57]

Equation [2.53] is now used to introduce the variables B and C into the Poynting vector.
If the magnitude of the electric field in the final material is denoted by E21,, then the

Poynting vector in material 0 can be shown to be:

S0 = g; RelBEzb '(CEZb )* ) r [258]
= éRe(BC* )Ezzb ' r

The Poynting vector in the final material is:

27

C

§=
287x

N2E§b -r [2.59]

Equation [2.57] can be thought of as proportional to the energy entering the layer, while
equation [2.59] is proportional to the energy leaving the layer. To calculate the
transmission through the layer, reflection of the incident power must also be considered.

Let Pm be the total incident power. This leads to the expression:
S0 = (1 — R)P,.,, [2.60]

Solving for PM gives:

in T [2.61]

Equation [2.59] is equivalent to the power leaving the layer, PM, so from [2.58] and

[2.61], the transmission through the layer can be easily calculated as:

 

 

[2.62]

where R can be calculated from equation [2.56]. Although transmission calculations can
be made from equation [2.62], this expression can be further simplified in an effort to aid

computational speed. First, combining equations [2.54] and [2.56] gives:

 

 

N,.9 N,.9
R_ B , B
N +9 N +2
o B o B [2.63]

_ NOB—C . NOB—C
NOB+C NOB+C

Using [2.63], l-R can be seen to be:

28

1—R- NOB+C . NOB+C _ NOB—C . NOB—C
NOB+C NOB+C NOB+C NOB+C
2N0[BC* + B‘C)
(NOB + CXNOB + C)*

 

Equation [2.64], can be inserted into [2.62] to give an expression for the transmission

through the film:

T = 4N°N2 ,, [2.65]
(NOB + CXNOB + C)

 

Equation [2.65] is used to calculate transmission when applying MacLeod’s technique.
However, as derived, equation [2.65] is specifically for one layer. This is not a problem,

though, since this approach can easily be extended to the case of multiple layers.

If a two-layered system is considered, as shown in Figure 2.6, a development analogous

to the preceding will give the result:

1 L - ._'L - 1
r x E00 [y] ___ cos 51 N1 srn 61 cos 52 N2 srn 62 [N :lr XE” [2.66]
iNl sin 61 cos 61 iNz sin 62 cos 62 3

where materials are now indexed by 0 through 3.

29

material 0

 

 

material 2

 

 

 

material 3

O O O
\/\ d1 material 1
\/\. e 0 (12

Figure 2.6 A multilayer stack of materials
For a stack of an arbitrary number of m layers, such an analysis gives
1'
1 m cos 5 — sin 5 1
rXEOa[Y:| = H n N,, n [N :ler(,,,+1)b [2.67]
"=1 iN,, sin 5,, cos 5,, "”1
The characteristic matrix for this assembly is written:
i
B m cos 5 — sin 5 1
"=1 iN,, sin 5,, cos 5,, "”1
Equation [2.65] can be used to calculate the transmission of the assembly, by simply

rewriting the equation:

T = 4N°Nm+1 [2.69]

(NOB + CXNOB + C)’

 

As an example, a modeling calculation for a gold-diamond-gold Fabry-Perot resonator is

shown in Figure 2.7. Calculations are based upon a diamond membrane thickness of 1.0

30

11m, and partially transparent gold film coatings of 25 nm. Optical data for gold is taken
from Kingslake”. The index of refraction for diamond is taken from the Sellmeier
equation. These modeling results account for absorption in the gold films but do not

account for absorption in the diamond or for scattering losses at rough surfaces.

0.7 I I I I I I

 

0.6 - .

.o .0
§ 01
I 7
1 1

Transmission
.0
OD
l
l

 

.9
N
r
r

 

 

 

 

 

 

 

 

 

11 12

1.3 1.4 1.5
Wawlength (microns)

Figure 2.7 Theoretical calculation of transmission versus wavelength for a gold-diamond-gold optical
resonator with d=1um of diamond, and 25nm gold layers on each side.

2.4.2 Transfer Matrix Method

The transfer matrix technique36 explicitly uses the Fresnel coefficients along with a

superposition of electric fields to solve the problem of multilayer optics.

31

 

 

 

 

 

 

A
130+ E0- no
a
i
E1+ E]. E d m
i
v i b
E2+ “2
V
Figure 2.8 A single thin layer

Figure 2.8 shows a thin layer, of thickness d, where a and b denote the two material
interfaces in the system. An electric field of E0,+ is incident upon the structure. The
incident field causes a reflected field, E03} which travels in the opposite direction to the
incident field, as well as fields internal to the layer, Em+ and Ela', where the + and - signs
indicate direction of travel. The field transmitted through the layer is E213. It is assumed
that E21; is equal to zero, which means that there is no optical source or additional

material boundaries to cause reflection in medium n2.

The aim of the transfer matrix method is to find the ratio of the reflected, Bog, (or
transmitted, E213) field to the incident field, E0; This is accomplished by noting that at
the interface a, E]: is the sum of the fraction of E02,+ transmitted through the interface

and the fraction of E1; reflected from the interface, which can be written as:

E3, = iEf; JQEI; [2.70]

’01 ’01

32

Similarly, the reflected field can also be written as a sum of two other fields, which can

be written in matrix notation as:
— — +
an = ‘1oE1a + ’01an
Inserting equation [2.70] into equation [2.71] gives:

_ __ 1 + ’10 _.
E00 =t10Ela + ’01(—E1a __Ela
’01 ’01

_ r01 + 1 —
- —Ela + *(‘01’10 — '10’01)Era
’01 ’01

Equations [2.70] and [2.72] can be written in matrix form as:

Elia _i[1 "10 ]Ei;
56., ’01’01 [01’10‘70100 E1;

Equation [2.73] can be significantly simplified by noting that:

, , _, , =(2"0X2"1)_("0-"1X"1‘"0)
0110 0110 2
("0+"1) ("01""1)2
:4n0n1+ng+n12—2n0n1:1

("0+”1)2

 

and that:

00='br

So, incorporating equations [2.74] and [2.75] into equation [2.73] gives:

E00 _L[1 ’01] E12
Efia ‘01’01 1 E1;

[2.71]

[2.72]

[2.73]

[2.74]

[2.75]

[2.76]

Equation [2.76] is used to find the relationship between the fields on opposite sides of a

boundary. Equation [2.45] can be used to see how the fields propagate across the

thickness of the layer. Writing the electric fields of [2.45] in matrix form gives:

33

i5
El: = e O. E 17’ [2.77]
E; 0 ['5 El},

Equations [2.76] and [2.77] can be combined to describe the wave from material no up to

interface b in material 111:

+ 1 i5 +
5‘1“ =-l-[ r01] 6 9,; Elf 12.781
an ’01 ’01 1 0 e 1 E1},

The description of the system is complete when the matrix transferring the wave from

material n1 into material n2 is written:

Eli) =i[1 011557) [2.79]
51—1, ’12 ’12 0 0

Equations [2.78] and [2.79] are then combined to write a full description of the wave

from material no to material n2:

1:3,, 1 1 re. .16 o 1 0 52+,
_ =— as [2.80]
500 ’01'12 ’01 1 0 e ’12 0 0

which has a general form of:
+ b +
59:1 = [a ] E21) [2.81]
EOa C d O

The ratio of the transmitted field to the incident field can then be found by:

51-1

+ [2.82]
E00 a

The power transmission can be calculated by:
T =[l) [l] [2.83]
a a

34

This approach can be easily generalized to the case or many layers. Essentially, one
matrix is needed for each interface, and an additional matrix is needed for each layer to
account for the change in phase as the wave travels the thickness of the layer. Thus, for a

system of m layers between interfaces a and b, the equivalent of equation [2.80] would

[53,]: (El) 1 [ 1 rj><j+11][e‘°‘if 06 ]
E60 j=0 t(j)(j+l)t(j+l)(j+2) 'lJ'XJ'H) 1 O e—l I [2.84]

.[ 1 ”[5610;
’(meH) 0 0

Despite it’s more complex appearance, equation [2.84] still can be reduced to a general

be:

 

form of equation [2.81], and thus equation [2.82] and [2.83] can be used to find

transmission for the system.

2.5 Effect of Interface Roughness on Transmission Through a Single Surface

When a wave reflects from a smooth surface, the reflection is in a specific direction,
according to electromagnetic theory. This is called the specular direction. When a wave
is incident on a medium with a small, random surface roughness, the reflected energy will
be distributed in some manner about the specular direction. This research will focus on a
one-dimensional transmission problem, so when the reflected energy is no longer totally

in the specular direction, it corresponds to a loss mechanism.

Figure 2.9 shows the general scattering geometry for a harmonic plane wave incident

upon a rough surface with a mean value of 2:0 for the surface heights, where I is the

35

plane of incidence, which is perpendicular to the y direction, and R is the plane of
reflection which can be rotated by an angle 03 away from the x-axis. The incident beam
is rotated by an angle 01 from the z-axis, while the reflected beam is rotated by an angle
02 from the z-axis. Note that for a smooth surface and specular reflection, the angle 02

will be equal to 0., and 03 will be zero.
[ z

91 y
92

 

 

R
)93

Figure 2.9 The general scattering geometry

Beckman37 shows that the case of a rough surface with a random Gaussian distribution,
which has a small standard deviation compared to the incident wavelength, can be solved
with only two dimensions if the parameters of the distribution are constant over the x-y

plane. Under these conditions, 03 is nearly equal to zero and 02 is nearly equal to 01.

36

Additional assumptions that Beckman employs are: that the incident wavelength, A, is
much larger than the RMS value of the surface roughness, 0‘, that the extent of the
surface in the x and y directions, L, is much larger than xi; and that no correlation exists in
the distribution. Conceptually, this set of assumptions may be likened to a radio wave
being incident upon a dense forest with treetops spaced much closer than the incident
wavelength, thus the treetops would form a very random surface profile. This is in
contrast to a gently rolling meadow, where the surface height could still be considered

random, but the separation between peaks is not small compared to the wavelength.

Under this set of assumptions, Beckman shows that the mean field reflected in the

specular direction can be written as:

 

(E20) = E2 sincl:-2—f— (sin 61 — sin 62M] - exp{- 2:32 (00361 + cos 92 )2] [2.85]

Where E2 is the total field reflected, and E20 is the field reflected in the specular direction.
For normal incidence, as is the case for this research, 01 and 02 are taken to be zero, and

the reflected power in the specular direction is then shown to be:

2
(E20)(EZO) = £2.15; exp[-(i§g) ] [2.86]

This result can also be obtained by considering a sum of plane waves.

37

1F M

111

 

 

 

Figure 2.10 Cross-section of a rough interface

Figure 2.10 shows two beams incident upon a rough surface. RA and R3 are the reflected
beams, and Ah is the difference in surface height from which the beams are reflected.
Assurrring that the incident beams are initially in-phase, of the same amplitude, and that

they are normally incident on the sample, the incident wave can be written as:

1.27010 Z 1.27020 Z 1.270202

E,0(z)=2E,-e ’10 =E,-e "0 +E,-e "0 [2.87]

 

where ’10 is the free space wavelength and no is the index of refraction. The reflected

wave can then be written as:

-i—2’”’0 (z+2Ah) —i 2”"0 z

E,0(z)=roE,-e *0 +rOE,-e "0 [2.88]

 

where r0 is the Fresnel reflection coefficient for the surface”:

 

r0 = "1—"0 [2.89]
n1 ‘1' ’10

38

The phase difference between the reflected waves is:

2
R, = R, exp {44%) [2.90]

The reflected wave can be rewritten as:

2 4
7010 Z -i 7010A}!

 

-i

E,0(z)=r0E,-e ’10 1+e ’10 [2.91]

If instead of having reflections from two surfaces, reflections from many surfaces are

considered, the reflected wave could be written as:

 

 

 

 

42:)” z 42/170100 (2+2Ah1) 423:0 (2+2Ah2)
ErO (z) = rOE,e + rOE,e + rOE,e + [2.92]
.27mo(
-t z+2Ahj)
E,0(z) = rOE,Ze ‘0 [2.93]
1'

If the summation is converted to an integration over the z-direction, and the surface
heights are assumed to have a Gaussian distribution in the z—direction, denoted by z’, the

average reflected field will be:

 

 

<E,0(z))=rOE,-e ’10 [w(z')e 10 dz' [2.94]

 

w(z')= 1 e 202 [2.95]

Inserting w(z’) into the integral of equation [2.95], the reflected field can be found to be:

39

 

-i

2
271710 z _1[ 470100]

 

(E,0(z)> = rOE,e ’10 e 2 ’10 [2.96]
The total power reflected in the specular direction can be found as:
_,26.0 Z _[[4m06]2 ,mz _1[47m00‘]2
(Er0(Z»<Er0(Z»* = ’oEre '10 e ’10 ’oEi*e ’10 e ’10 [297]
{MY
(Er0(z»<ErO (Z))* = rOZEiE:e 10 [2'98]

This is the same as Beckman’s result for normal incidence.

Filinski39 first shows the same result, and then extends this approach to transmission

through the interface. Consider the situation shown in Figure 2.11.

 

 

A B
R n
A RB 0
f\ \I... f
T
TA B m

 

 

 

Figure 2.11 Cross-section of a rough interface

40

A and B represent incident beams of light. RA and R3 represent reflected beams, and TA
and To represent transmitted beams. Ah is the difference in height between the points on
the surface the beams are reflected from. The two different media are again represented
by their indices of refraction, no and n1. Assuming that A and B are initially in phase and
normally incident upon the surface, the phase difference of the transmitted waves can be

expressed as:

ACIJT, : 27rAh

 

(n1 — 710) [2.99]

The transmitted wave could thus be written as:

 

 

.27011 .2721}:
-1 z -1 (n1 -—n0)
E,0(z) = tOE,e ’10 1+ e "0 [2.100]
where to is the transmission coefficient for the surface:
2
to = "0 [2.101]

 

n, + "0
If, as in equation [2.93], a summation of many interfaces is considered the transmitted
wave can be written as:
4271”1 z -i 2m (111—no)
I10

E.o(z)=toE,-e *0 2e
1

 

 

[2.102]

Again, this summation will be converted to an integration over a Gaussian distribution of

surface heights:

 

(E,0(z))=t0E,-e *0 [w(z')e "0 dz' [2.103]

41

w(z’) is again the Gaussian distribution with a standard of 0', and thus the integral can be

evaluated to give:

,27011 1 [2fl0(n1~rro))2
(Et0(z)>=tOEie 10 e

-5 [,0

[2.104]

This can now be used to find the power transmission coefficient for the rough surface:

T :|<Er0(z»<Et0(Z»*nl

 

 

[2.105]

which Filinski expresses as:

 

T, = T, exp[— (ACDT, )2 j = (1 — R, )ex — [ 27mg: - "0 )) [2.106]

where T. is the transmission if the surface was smooth, and R, is the power reflection

coefficient for a smooth surface.

2.6 Concluding Remarks

In this chapter, the optical properties of diamond, including thin film diamond of the type
used in this research have been reviewed. The matrix approach to multilayer modeling of
thin films was described, as well as performance measures of Fabry-Perot resonators and

impacts of surface roughness. The research presented in the remainder of this thesis aims
to incorporate all of these issues in the fabrication and analysis of on—chip diamond-based

Fabry-Perot resonators.

42

References:

1 J. Verdeyor, Laser Electronics, Prentice Hall, Englewood Cliffs, NJ, 1989.

2 J. Han and D. P. Neikirk, "Deflection behavior of Fabry-Perot pressure sensors having
planar and corrugated membrane," SPIE's Micromachining and Microfabrication '96
Symposium: Micromachined Devices and Components H, R. Roop and K. Chau, Proc.
SPIE 2882, Austin, Texas, USA, 14-15 October, 1996, pp. 79-90.

3 S. Ahmadi and M. Zaghloul, “A Fabry-Perot Optical Sensor System-on-Chip”, Can. J.
Elect. & Comp. Eng, Vol. 26, NO 3/4, July/Oct 2001, PP159-162.

4 R. A. Booth and D. K. Reinhard, “Diamond Thin-Film Fabry-Perot Optical
Resonators”, ADC-PCT 2001, August 5-10 2001, Auburn, AL, USA.

5 J. H. Correia and C. Couto, “MEMS: a New Joker in Microinstrumentation”, IEEE
Industrial Electronics Newsletter, January 2000.

6 J. H. Correia, M. Bartek and R. F. Wolffenbuttel, “High Selectivity single-chip
spectrometer for operation at visible wavelengths”, Proc. 1998 IEEE IEDM, December 6-
9, 1998, San Francisco, USA.

7 J. H. Correia, M. Bartek, R. F. Wolffenbuttel, “Hi gh-Selectivity Single—Chip
Spectrometer in Silicon for Operation at Visible Part of the Spectrum”, IEEE
Transactions on Electron Devices, Vol. 47, No. 3, March 2000, PP 553-559.

8 M. Bartek, I Novotny, J. H. Correia, V. Tvarozek, “Quality Factor of Thin-Film Fabry-
Perot Resonators: Dependence on Interface Roughness”, EuroSensors XIII, September
12-15, 1999, The Hague, The Netherlands.

9 M. Bartek, J. H. Correia, S. H. Kong, and R. F. Wolffenbuttel, “Stray-Light
Compensation in Thin-Film Fabry-Perot Optical Filters: Application to an On-Chip
Spectrometer”, Transducers ’99, June 7-10, 1999, Sendai, Japan.

10 J .H. Jerman, D.J. Clift and SR. Mallisnson, "A rrriniature Fabry-Perot Interferometer
with a Corrugated Diaphragm Support," Technical Digest IEEE Solid State Sensor and
Actuator Workshop, Hilton Head, June 1990, PP 140-144.

11 H. Saari, R. Mannila, J. Antila, M Blomberg, O. Rusaned, “Miniaturized Gas Sensor
Using a Micromachined Fabry-Perot Interferometer”, Preparing for the Future, Vol. 10
No. 3, October 2000 PP 4-5.

‘2 Y. Kim and D. P. Neikirk, "Design for Manufacture of Micro Fabry-Perot Cavity-
based Sensors" Sensors and Actuators A 50, Jan. 1996, pp. 141-146.

43

13 Y. Kim and D. P. Neikirk, "Micromachined Fabry-Perot Cavity Pressure Transducer"
IEEE Photonics Technology Letters 7, Dec. 1995, pp. 1471-1473.

‘4 D. Shin, S. Tibuleac, T. A. Maldonado, and R. Magnusson, “Thin-Film optical filters
with diffractive elements and waveguides”, Opt. Eng, September 1998, PP 2634-2646.

15 M. Seal and W. J. P. van Enckevort, “Applications of diamond in optics”, SPIE Vol.
969 Diamond Optics, 1988 PP 144-152.

'6 F. Urbach, Phys. Rev., 92, PP1324, 1953.

17 C. D. Clark, “Optical Properties of Natural Diamond”, Physical Properties of Diamond,
ed. R. Berman, Clarendon Press, Oxford, 1965.

'8 D. K. Reinhard, Diamond Films Handbook, Chapter 14, J. Asmussen and D. K.
Reinhard, Eds, Marcel Dekker, Inc. New York, NY, 2001.

19 M. Thomas and W. J. Tropf, Johns Hopkins APL Tech. Dig. 14; P.16, 1993.
2° M. P. Marder, Condensed Matter Physics, John Wiley & Sons, New York, NY, 2000.

2' A. Feldman and L. H. Robins, “Critical assessment of optical properties of CVD
Diamond Films”, Applications of Diamond Films and Related Materials, Y. Tzeng, M.
Yoshikawa, M. Murakawa, and A. Feldman, Eds, Elsevier Science, 1991, PP 181-188.

22 A. T. Collins, M. Kamo, Y. Sato, “A spectroscopic study of optical centers in diamond
grown by microwave-assisted chemical vapor deposition”, J. Mater. Res., 5, 1990, PP
2507-25 14.

23 V. Vorlicek, J. Rosa, M. Vanecek, M. Nesladek, L. M. Stals, “Quantitative study of
Rarnan scattering and defect optical absorption in CVD diamond films”, Diamond and
Related Materials, 6 (5-7): 704-707 1997.

2" D. C. Harris, “Development of Chemical-Vapor-Deposited Diamond for Infrared
Optical Applications. Status Report and Summary of Properties”, Naval Air Warfare
Center Weapons Division, July 1994.

25 A.J. Gatesman, R. H. Giles, and J. Waldman, Diamond Optics HI, SPIE Volume. 1325,
The International Society for Optical Engineering, Washington, 1990.

2” L.H. Robbins, E. N. Farabaugh, A. Feldman, Diamond Optics IV. SPIE Volume 1534,
The International Society for Optical Engineering, Washington, 1991.

2’ x. Ying, Y. Shen, H. Xue, L. Jianhai, z. Xing, J. Xu and F. Zhang, Diamond Optics V,
SPIE Vol. 1759, The International Society for Optical Engineering, Washington, 1992.

44

28 P. Rouard, Annuel de Physique 7, 291, (1937).

29 A. Nussbaum and R. A. Phillips, Contemporary Optics For Scientists and Engrjreers,
Prentice Hall, Inc., New Jersey, 1976.

30 O. S. Heavens, Optical Properties of Thin Solid Films, Butterworths, London, 1955.

31 F. Abeles, Advanced Optical Techniques, A.C.S. Van Heel ed, North-Holland
Publishing Co., Amsterdam, 1967.

32 HA. MacLeod, Applied Optics and Optical Engineering, R.R. Shannon and J .C.
Wyant eds, Academic Press, New York, NY, 1987.

33 HA. MacLeod, Thin-Film Optical Filters, American Elsevier Publishing Co., New
York, NY, 1969.

3" Born and Wolf, Principles of Optics, Pergamon Press, New York, NY, 1959.

35 R. Kingslake, Applied Optics and Optical Engineering, Academic Press, New York,
NY, 1965.

36 0.8. Heavens, Optical properties of Thin Solid Films, Dover, New York, 1965.

37 P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from
Rough Surfaces, Pergamon Press, Oxford, 1963.

38 R. Harrington, Time-Harmonic Electromagnetic Fields, McGraw Hill, New York, NY
1961.

391. Filinski, “Effects of Sample Imperfections on Optical Spectra”, Phys. Stat. Sol. (b),
49, 577-588, (1972).

45

Chapter 3: Experimental Methods

3.1 Introduction

This chapter describes the experimental aspect of this research, including the fabrication
of the device and the techniques used to characterize the device optically. The goal of
this chapter is to convey an understanding of the methods used to create and measure the

samples in this research.

3.2.Device Fabrication

The fabrication of the device involves a series of many steps. First, an overview of the
fabrication sequence is given. Next, the oxidation and photolithography steps are
discussed. The diamond deposition is described, and then the method used to achieve
through-etch of the wafer is presented. At this point in the sequence it is necessary to
perform an optical measurement of the sample to document the diamond thickness before
proceeding to the last fabrication sequence, sputtering thin gold films to complete the
Fabry-Perot device. Also, atomic force microscopy (AFM) and scanning electron
microscopy (SEM) are used to document device topology and morphology. Finally the

methods used to measure the optical performance of the finished device are described.

46

3.2.1 Fabrication Overview

Figure 3.1 shows a conceptual cross-sectional final view of the device constructed for this
study. The Fabry-Perot device is fabricated on a double-side polished, 2" diameter (100)
lightly doped (1-10 ohm-cm) p-type silicon wafer with an approximate thickness of 250

pm, which has been through an RCA cleaning process].

As can be seen in Figure 3.1, in certain spots the entire thickness of the wafer is
anisotropically etched away in order to form a diamond membrane approximately 1pm
thick, supported on 4 sides by the remaining silicon. The first step in the device
fabrication is to grow a thermal oxide on the wafer that will eventually serve as a mask
for the aforementioned through-etch of the wafer. For this reason, the desired thickness
of the oxide layer is a function of the wafer thickness, as well as the relative etch rates of

silicon and silicon-dioxide in the etching solution.

 

 

 

 

 

 

 

 

“an... -4... ..C..-‘. ... ‘W \K .x. a. .7 4 s ....x...m1_
‘ Q
E _ =--.==a=.=l; .. ‘1

 

 

E] Silicon Gold
E] SiOz [3 Diamond

Figure 3.1 Cross-sectional view of two Fabry-Perot resonators. Drawing not to scale.

Bean2 gives the oxide etch rate in KOH/H20 solution as approximately 3 nm per minute.

Others report even slower rates for etching oxide. For the KOH/1120 solution used in this

47

research, the reported etch rates for silicon are between 250 and 330nm per minute. This
means the minimum oxide mask thickness should be approximately 1% of the thickness
of the wafer in order to avoid the etching of silicon in the masked area. However, in this
application, some etching of the masked silicon is tolerable since the silicon would still
be thick enough to support the diamond windows. Thus, for a 250m thick silicon wafer,
the target oxide thickness was taken to be 211m. After oxidation, the next step is
patterning of the oxide followed by diamond deposition and silicon etching. Finally, gold

rrrirrors are sputtered, resulting in the structure in Figure 3.1.

3.2.2 Oxidation

For a target oxide thickness of approximately 211m, the Deal-Grove oxidation model3
indicates approximately 8.5 hours of wet oxidation, or 150 hours of dry oxidation time, at
1100°C. However, the ideal, 100% wet oxidation considered in the model can be difficult
to achieve in the laboratory without considerable effort. Recognizing that 8.5 hours
would be insufficient, the wafers were oxidized for approximately 20 hours at 1100°C
under wet oxidation conditions. The weight of the wafer was measured before and after
the oxidation. The weight gained during the oxidation can be used to estimate the

resulting oxide thickness.

To calculate the oxide thickness via weight gain measurements, it is assumed that the
total number of silicon atoms in the wafer does not change, and that all of the weight gain

comes from oxygen atoms bonding with the silicon in the wafer to form an 8102 layer.

48

This means that the weight of the SiOz on the wafer, wox, is related to the measured

weight gain, wg, by:

28.09
w = 1+—— -w 3.1
0" l 2.16.00l g l 1

Where 28.09 and 16.00 are the atomic masses of silicon and oxygen, respectively.
Additionally, it is assumed that the wafer is a perfect 2-inch diameter circle and that
oxide growth on the edges can be neglected. This means that the thickness of the oxide
layer on one side of the wafer, tax, can be determined by the using the density of SiOz,

pox, and Wax-

wox [3.2]

t = 2
2'pox°7z"r

 

ox

The factor of two in the denominator comes from the fact that the oxide is grown on both
sides of the wafer. The radius of the wafer is r. If pm, is 2.2 grams per cubic centimeter,
and r is 2.54cm, equations [3.1] and [3.2] can be combined and solved to give a

relationship between w,, and tax:

1.878-w
to, = ——“— = 0.02106- w, [3.3]
89.18

where the units of to, are in cm, and w, is in grams. Expressed in units of um and grams,
equation [3.3] is:

to, = 210.6-wg [3.4]

As an 8102 layer is grown on a silicon surface, a certain thickness of the silicon is lost. A
common rule of thumb is that the thickness of silicon consumed by the oxidation is 44%
of the thickness of the final resulting oxide layer. This means that after the oxidation, the

thickness of the wafer, tw, is approximately:

49

2,, = 250— 2044-110, = 250-1853 w, [3.5]

where the factor of two accounts for the fact the silicon is consumed from both surfaces.
Again, the units for t... are um, and w, is in grams, and a starting thickness for the wafer

of 250m is assumed.

A Thermco Mini Brute diffusion furnace was used to thermally oxidize the wafer. A
heated bubbler containing deionized (DI) water was used to provide steam for the
oxidation. Grade 5.0 oxygen (meaning “five-nines”, or, 99.999% pure) was pumped
through the bubbler and into the furnace at an approximate rate of 200 standard cubic
centimeters per minute (sccm). MSU has recently acquired a PECVD system which can
deposit oxides and nitrides at lower deposition temperatures, which in the future may be a
better method of producing a masking layer for the silicon through etch due to both

thermal budget considerations, as well as the possibility of an increased deposition rate.

Weight gain measurements showed that these growth conditions resulted in an oxide
layer approximately l.5p.m thick. This is slightly less than the target oxide thickness, but

was found adequate for masking purposes. As shown in Figure 3.2, the oxide was grown

on both sides of the wafer.

50

 

. fl . . . .;. . . . .j. ., ;. . . ... ._. ;. . . . . .j. . . . . . . . -. , . . . . . . . . . . ., '. ., . . . . . 1. . ..............................

 

 

 

 

 

Oxide D Silicon

 

Figure 3.2 Cross-sectional view of the wafer after oxidation. Drawing not to scale.

3.2.3 Photolithography

After oxidation, one side of the wafer is patterned with an array of squares, each square
measuring approximately 2mm by 2mm. The squares are aligned with the flat edge of

the wafer in order to align the pattern with the atomic lattice of the silicon wafer.

The pattern on the wafer is created with Waycoat HR200 negative photoresist. Before
the resist is applied to the wafer, the wafer is cleaned by spraying it with acetone for
approximately 30 seconds, followed by a methanol spray for approximately 30 seconds, a
DI water rinse for 2 rrrinutes, and then blown dry by N2. The wafer is then placed in a
convection oven for a “bake-out” step at 200°C for 30 rrrinutes to assure the surface is dry
for good resist adhesion. After the bake-out, the wafer is placed on the resist spinner,
coated with resist, and spun at 2000 rpms for 30 seconds. This corresponds to a pre-bake
resist thickness of approximately 111m. The coated wafer is then placed in a convection

oven for a “pro-bake” procedure, which occurs at 65°C for 20 minutes.

51

After the pre-bake, the wafer is ready for exposure. A Karl Suss MJB-3 mask aligner
equipped with a UV300 optical source was used. This optical source exposes the
photoresist with wavelengths from 280 to 350nm. Exposure is in the soft-contact mode
for approximately 10 seconds. The MJB-3 is capable of very high resolution (up to 0.4
microns in this configuration), but given the large feature sizes of the device constructed

for this research, such high resolutions are not necessary in this case.

After exposure, the wafer is placed directly in the Waycoat negative photoresist
developer solution. The wafer is agitated in this solution for 100 seconds. After the
developer, the wafer is transferred to a beaker of xylene and agitated for approximately
20 seconds. The wafer is then placed in a beaker of isopropyl alcohol and again agitated
for approximately 20 seconds, and then dried with blowing N2. The wafer is next placed
in a convection oven for a “post-bake” procedure, which occurs at 135°C for 15 minutes.

After the post-bake, the wafer and pattern are ready for the oxide etch.

3.2.4 Oxide Etch

An oxide etch is performed after the photolithography process. The goal of the etching
step is to transfer the pattern in the photoresist to the oxide, while also removing all of the
oxide from the unpattemed side of the wafer. The etch solution was a 6:1 mixture of
40% W to 50% HF at room temperature. This solution etches the oxide at an
approximate rate of 100nm per minute, so the etching time of 15 rrrinutes was expected
for the 1.5um oxide layer. Since 8102 is hydrophilic and Si is not, the progress of the

etch can be monitored by periodically removing the wafer from the etching solution,

52

submerging it in a beaker of water, and then removing the wafer and watching the
behavior of the wafer as it streams off the wafer. When the oxide has been removed, the
water will run off of the silicon rapidly and essentially completely, while when the wafer
is still coated with oxide the water will ‘sheet’ and cling to the wafer. The oxide etch was
terminated when observing the water indicated that the S102 had been removed. Since
the oxide etch solution is hazardous, after terminating the etch in a beaker of de-ionized
(DI) water, the wafer was transferred to a second beaker of DI water to dilute any
remaining etchant, next rinsed in running DI water for two minutes, and then dried with

blowing N2.

After the oxide etch, the photoresist has served its purpose and must be removed. A
beaker containing Waycoat Microstrip photoresist remover is heated to 85°C on a
hotplate. The wafer is placed in the heated solution for 10 minutes. The wafer is then
sprayed with methanol for approximately 20 seconds, and then rinsed in running DI water
for approximately 2 minutes. Finally, the wafer is blown dry with N 2. This procedure
was used since the wafers for this research were processed in batches of 1 or 2 at a time.
If larger batches of wafers were to be processed, a two-step procedure for stripping the

photoresist may be appropriate.

Figure 3.3 shows a cross-sectional view of the wafer after etching the oxide and

removing the photoresist.

53

 

   

 

 

 

 

Oxide [:l Silicon

 

Figure 3.3 Cross section of the wafer after oxide etch and photoresist removal. Drawing not to scale.

3.2.5 Diamond Deposition

Various techniques for diamond deposition were explored at the beginning of this project.
This included experiments in seeding, deposition chemistry, power and chamber
configurations, and wafer sizes. What is described in this section was the procedure that
resulted in the best optical properties of the diamond film and the resulting free-standing

windows and Fabry-Perot resonators.

After etching and photoresist removal, one side of the wafer is bare silicon, while the

other side has a patterned oxide. Figure 3.4 shows a photograph of the patterned oxide.

54

 

Figure 3.4 Patterned oxide on 2-inch diameter wafer

The wafer is then polish seeded for diamond film growth on the non-pattemed side, and a
diamond film is then deposited on that side of the wafer. The method used to seed the
wafer was based on a method originally suggested by Windischmann, and later modified
by Ulczynski". Briefly, the wafer is cleaned in acetone, methanol, and then deionized
water for 2 minutes each step. The wafer is then dried with blowing N2. Next, the wafer
is placed on a KimWipe and lightly dusted with diamond powder, which is then used to
polish the wafer by hand for several minutes. The polishing procedure is to use a
KimWipe to hold the wafer steady on the tabletop, while wrapping the index finger of the

other hand with a KimWipe and using the index finger to wipe the diamond powder

55

across the surface of the wafer in a circular pattern for several minutes. Gloves should be

worn at all times while seeding the wafer.

Both synthetic and natural diamond powders, all sourced from Amplex, were used for
seeding, both showing good results, although the synthetic powder may have been more
consistent in producing films that exhibit good optical properties. Both 0.25pm and
0.10pm maximum particle size diamond powders were used in natural and synthetic
varieties, with no obvious advantage being seen with either size. When the polishing step
was complete, the wafer was carefully wiped clean of any blemishes or marks visible to
the eye, and again blown with N2 to whisk away any small particles clinging to the wafer.
This method differs from Ulczynski in that a final cleaning step has been omitted. The
omission of the cleaning step seemed to have little impact on the wafers for purposes of

this research, and shortened the seeding procedure significantly.

The diamond films were deposited using a Microwave Cavity Plasma Reactor (MCPR)
of the Asmussen design, built at Michigan State Universitys. The particular system used
in this research was developed to deposit diamond at relatively low temperatures, which
allows films to be deposited on a wide array of substrates which may otherwise prove
difficult to coat with diamond. The MCPR was configured in an end-feed style for this
research. Ulcyznski4 refers to this configuration as MCPR Configuration 7, and also
details other configurations for this reactor, as well as giving an overview of various
deposition conditions and results. The main variable in MCPR Configuration 7 is the

height of a quartz ring which supports the substrate holder. For this research, a quartz

56

ring 59mm tall was used. The two user adjustable dimensions of the MCPR are the
cavity length and the length that the probe protrudes into the cavity. For this work,
typical cavity lengths were on the order of 21.5cm, and the typical probe insertion depths
were on the order of 2.5cm. These values vary slightly depending on the plasma pressure
and other deposition variables. These parameters are selected to minimize the reflected
microwave power during the deposition. Figure 3.5 shows a cross-section of the wafer

after the diamond deposition.

 

   

 

 

 

 

 

Oxide l:l Silicon [:1 Diamond

 

Figure 3.5 Cross section of wafer after diamond deposition. Drawing not to scale.

Diamond deposition parameters prove to be very critical not only to optical quality, but
film integrity as well. Intrinsic stress in the film due to thermal mismatch is a large factor
in the flatness and durability of the film". The deposition was performed at 35 Torr, at a
substrate temperature of approximately 660°C. This resulted in a flat membrane when
the underlying silicon was etched away. Depositing at substantially lower temperatures
and pressures would usually result in films that would wrinkle due to compressive stress,
and in some cases break, at the completion of the subsequent silicon through-etch. Figure

3.6 shows a picture taken through an optical microscope of a window that wrinkled upon

57

completion of the through-etch, due to depositing the diamond at too low of a substrate

temperature.

 

Figure 3.6 Wrinkled window resulting from low deposition temperature. This is an optical image

taken with dark-field illumination.

At higher deposition temperatures, the windows would remain flat after the silicon had
been etched away. Figure 3.7 shows a flat window taken under the same dark-field
illumination as Figure 3.6. Since the window in Figure 3.7 is very flat compared to the

window in Figure 3.6, the dark-field image is not very bright in this case.

 

Figure 3.7 Flat window shown for comparison to Figure 3.6. This is an optical image taken with

dark-field illumination.

58

An important parameter to the optical properties of the diamond film is the gas mixture
used in the plasma. All of the working devices in this research were grown with 200
sccm of H2, 8 sccm of CO2, and 3 sccm of CH4. This chemistry was found to produce
windows which were very transparent, although the presence of CO2 may lead to the

slower growth rate than could otherwise be attained.

At the conditions used for this research, the thickness of the diamond film is observed to
be non-uniform across the wafer as evidenced by the presence of many fringes in the
film. However, a rough estimate of the film thickness, to, can be made by using the
weight gained during the deposition, wd, in an equation similar to [3.2]:

W

pd - 7r ' r
where r is the diameter of the wafer, and pd is the density of diamond. Using 3.51 grams
per cubic centimeter for the density of diamond, and 2.54cm for the radius of the wafer,
and changing the length units to micrometers, equation 3.6 can be rewritten as:
td =140.6-wd [3.7]

where to is again in units of micrometers, and wd is in grams.

59

3.2.6 Silicon Through—Etch

After the diamond film deposition, the silicon wafer is through-etched using KOH to

create the diamond membranes. Figure 3.8 shows a cross section of the wafer after the

  ? /.\

Oxide l:l Silicon l: Diamond

silicon through etch.

 

 

   

 

 

 

 

 

Figure 3.8 Cross section of the wafer after the silicon etch. Drawing not to scale.

The patterned oxide is used as a mask, while the diamond film has proven to be resilient
to the KOH etch, thus serving as a mask on the second side of the wafer. The through-
etch is performed at a temperature of 60°C, in a 44/56 weight percentage KOH/H2O

solution7.

. 2’ 9 I
The literature 6 8 9

reports that the 44/56 solution etches (100) silicon at approximately
300 nm per minute. This corresponds to 18 um per hour, for a total etching time of about
14 hours for a 250 pm thick wafer. In practice, it was observed that the etch would not
start immediately, and proceeded slowly in the beginning, possibly due to the existence

of a native oxide on the wafer, and the possibility of material left on the backside of the

wafer from the diamond deposition. Actual etching times were around 18-20 hours to

60

etch completely through the wafer under these conditions. Another possible reason for
the etch taking longer than expected is that the concentration of the KOH solution may
not be constant for the duration of the etch. This phenomena is discussed briefly in a

subsequent paragraph, along with a possible solution.

Figure 3.9 shows a schematic of the apparatus constructed for the KOH etch.

Wafer Basket

False
/ bottom

/ Magnetic

stir bar

 

Teflon z

Beaker

 

  
 
 

 

 

 

Glass
Beaker

    
  

 

 

 

  
 

KOH/H2O
mixture @ 60°C

  

I Hot Plate I

Figure 3.9 Schematic of the apparatus used for KOH etching of silicon.

61

The apparatus of Figure 3.9 takes into account several problems associated with the KOH
etch. First, the etch must be maintained at approximately 60°C, and it must be contained
in a plastic beaker, as it will slowly etch any sort of glass container. However, the plastic
beaker cannot be placed on a hot plate as the temperature of the hot plate may become
high enough to melt the plastic. Thus, the large water bath is used to assure that the
plastic beaker will not see temperatures above 100°C. The Teflon beakers used in this
research were rated to 121°C. It was observed that the KOH would gradually precipitate
out of the mixture, and thus continuous stirring of the mixture is necessary. A standard
plastic-coated magnetic stir bar was employed, but it necessitated a change to the way in
which the wafer was held in the mixture. To this end, a false bottom was created in a
plastic wafer processing basket, such that the stir bar could be located safely beneath the
wafer. Once this apparatus was in place, the temperature of the etch was monitored with
respect to the temperature control knob on the hot plate until a suitable setting was found.
This corresponded to a water bath temperature of roughly 85°C, which lead to the KOH

solution having the intended 60°C operating point.

The apparatus can be improved from the configuration employed in this research.
Namely, a reflux condenser can be used to reduce the effects of evaporation on the
solution. The solution can be observed to evaporate approximately 200mL over the

course of the Si through-etch, which means that the 44/56 KOH/H2O content of the

mixture was not likely constant throughout the etching step.

62

When the diamond membrane is created by removing the silicon beneath it, the through
transmission of the membrane is measured as described in the Optical Measurements
section. The measurement step allows characterization of the particular window, namely
the thickness of the film at that point, as well as the surface roughness. Figures 3.10 and
3.11 show photographs of clear, flat diamond windows at this phase of the fabrication

sequence.

 

Figure 3.10 Looking through a clear diamond window onto a trans'stor (transistor shown for
purposes of illustration only).

63

 

Figure 3.11 Holes seen in wafer are actuauy flat, transparent diamond windows.

3.2.7 Gold Sputtering

Once the diamond membranes are formed and measured, the final step is to sputter coat
gold on both sides of the membrane. A commercial DC sputtering system used to
conductively coat samples for scanning electron microscopy was used to apply the gold
layer to the sample. A shadow mask is used to selectively coat the windows, instead of
coating the entire wafer at once. The diamond film tends to have a rougher surface on the
growth side as opposed to the silicon side of the film. The result is that the gold
deposition on the growth side of the membrane has to be slightly thicker to form a

continuous film. Experimentally, a ratio of 5/3 was determined for the thickness of the

gold film on the rough/smooth surfaces of the membrane for minimum gold thickness.

At this point, the device as shown in Figure 3.1 has been fabricated.

Typical thickness values for the gold films in this research were 18-35nm. Films thicker
than this range resulted in very low optical transmission, while films thinner than this
range were probably not continuous, and thus their optical properties did not follow

theory very well.

The thickness of the gold films can be estimated by one of two methods, depending upon
the sputtering system used. First, some sputtering systems are equipped with a film
thickness monitor, which directly provides some estimate of the thickness of the film
being applied. A second method employed was to place a microscope slide next the
diamond membrane in the sputtering system. Presumably, the thickness of the gold
deposited on the slide would be almost the same as that deposited on the diamond
membrane. The optical transmission through the slide can then be measured, and the

gold thickness mathematically determined by fitting a simulation to the measurement.

65

3.3 Optical Measurements

The device is characterized in two steps. First, before the diamond membrane is sputter
coated with gold, the through-transmission vs. wavelength is obtained for the diamond
membrane only. This allows independent determination of the thickness and surface
roughness of the particular window. Secondly, the transmission through the gold-coated

resonator is measured.

3.3.1 Measurement Procedure

The optical through-transmission measurement is performed by passing a monochromatic
beam of light through a particular diamond membrane or resonator, which is masked to
block any light from passing through the silicon or other diamond membranes. The
sample is placed in the system, and then carefully positioned for maximum power
transmission through the system. The wavelength is varied and the transmitted power
and the wavelength value at each wavelength of interest are recorded. The sample is then
removed from the optical system, and a mask with the same size opening as was used on
the sample is placed in the path of the beam. The monochromator is returned to the
wavelength noted during the positioning of the sample in the system, and again the
position of the mask is chosen for maximum power transmission through the system.
The power of light passing through the mask with no sample is then recorded for each
wavelength. The "sample" data is then normalized against the "air" data to obtain an

accurate value for percent transmission through the sample.

66

3.3.2 Measurement Apparatus

Three different monochromators were used during the course of this research to
document optical transmission as a function of wavelength: a Bausch & Lomb High
Intensity Grating Monochromator, a Beckman IR 4200 Infrared Spectrometer, and a

Perkin Elmer Lambda 9000 Monochromator.

Figure 3.12 shows a diagram of the experimental set-up used with the Bausch & Lomb
monochromator, which was used for the bulk of optical measurements. It consists of a
light source and monochromator, an adjustable aperture, two lenses, a sample holder, and
an optical power detector and meter. A Bausch & Lomb Tungsten Light Source is
coupled to a Bausch & Lomb High Intensity Monochromator, with an optical filter in the
middle to eliminate higher order spectra. Different monochromator and filter
combinations are available to span the visible wavelengths into the near IR. There are
two monochromator gratings used in this setup, called “VIS” and “IR-l”. Three filters
are employed, Corning C.S. Number 3.74 which cuts off wavelengths below 400nm,
Corning C.S. Number 2.56 which cuts off wavelengths below approximately 600nm, and
Coming C.S. Number 7.56 which cuts off wavelengths below 800nm. The VIS grating
used with the 3.74 filter spans wavelengths from 420nm — 760nm. The IRl grating
combined with the 2.56 filter goes from 700nm — 1050nm. The IR] grating combined

with the 7.56 filter spans the range of 1000nm to 1600nm.

The monochromator allows the user to change the entrance and exit slits, which is a

trade-off between how monochromatic the beam is, and the beam intensity. For this

67

research, the narrowest slit widths were selected, 0.75mm, giving the most
monochromatic beam available from this particular setup (4.8nm for the V13 grating and
9.6nm for the IR] grating), at the expense of some intensity. Even at the reduced beam

intensity, sufficient power reaches the detector to give good results.

The optical power transmitted through the sample is measured using 3 Newport 835
Optical Power Meter. Two different detectors are used in conjunction with this meter,
depending upon the wavelength being measured. The Newport 818-SC silicon detector is
used from the visible up to 1050nm in wavelength, and the 818-IR germanium detector is

used from 1000-1600nm.

To avoid the effects of stray light getting to the detector and offsetting the reading, the
measurement apparatus, from the monochromator to the optical power detector, is placed
under a cover during measurements, and the room lights are turned off. Additionally, the
apparatus is carefully checked to make sure that as much of the beam as possible is going
through the sample, and the remainder of the beam is blocked so that there are no

alternate paths to the detector.

The beam exiting the monochromator is first incident upon an adjustable aperture, which
is used to tailor the spot size of the beam to the physical dimensions of the diamond
membrane or resonator, as well as block any of the beam that would not be properly
incident upon the first lens in the system. The first lens focuses the beam to a small spot

so that almost the entire beam can pass through the small hole in the sample. The beam

68

is then incident upon a second lens, which focuses the beam onto the active area of the

detector.

 

 

 

 

 

 

 

 

 

 

 

 

Tungsten Light Source
[:1 Filter
Adjustable Monochromator
_l Aperture
Lens #1
Sample Holder/ Sample

C) Lens #2

 

Detector

\

 

 

 

 

 

 

Cover

Figure 3.12 Diagram of the optical measurement apparatus constructed for this research.

The two other spectrometers used for this research are both prefabricated, commercial
grade units. Both the Perkin Elmer and Beckman instruments are automated and offer
the user little direct control of sources, gratings, filters, or slit widths. Both instruments
perform dual beam measurements, and each instrument offers computerized data

collection.

69

The two commercial instruments have a large spot size compared to the dimensions of
the diamond windows constructed for this research. This can make getting good optical
transmission data difficult since a very large portion of the bearn’s power is blocked by

the mask, and thus the signal to noise ratio is increased accordingly.

3.4 Atomic Force Microscopy (AFM)

An AFM was used in this research to examine the surface of the diamond films. Data
from the AFM can give an indication of grain size, roughness, and the statistical nature of
the roughness. For this research, a Digital Instruments Nanoscope III AFM was used to
characterize certain samples. This instrument is capable of imaging atomic level features,
as well as operating as a Scanning Tunneling Microscope (STM). The instrument
requires that the sample be mounted on a lcm diameter disk, so for this research, the
samples must be broken into smaller pieces to be imaged by this AFM. STM mode

requires a conductive sample, and was not used for this research.

3.5 Scanning Electron Microscopy (SEM)

At certain points in this research, SEM images were used to evaluate fabrication
sequences. For this research, a Japan Electron Optics Laboratories (JEOL) 6400 SEM
was used. This particular SEM is equipped with a lanthanum hexaboride (LaB6) electron
source, and has a maximum magnification of 300,000. Such high magnifications are not

necessary for evaluation of the features found in this research, however. The SEM

70

requires that the sample be mounted on either a 1-inch diameter or 0.25-inch diameter
aluminum stub for insertion into the SEM. This means that the 2-inch wafer must be
broken into smaller pieces to examine with the SEM. Typically, non-conductive samples
must be gold coated for SEM imaging, although it was observed that for moderate
accelerating voltages of around 20kV or less, the samples for this research did not need to
be coated to produce acceptable images, but that coating made images somewhat easier to

obtain.

3.6 Summary

In this chapter, the fabrication sequence was explained and documented, as well as the
characterization steps. The through-etch of the wafer places some requirements on
earlier sequences, namely the lithography and oxidation. Additionally, the fabrication
sequence must be interrupted to characterize the device optically. Three methods for
optical characterization are described. Also, SEM and AFM methods for determining

morphology and evaluating fabrication sequences are discussed.

71

References:

l W. Kern and D. A. Puotinen, “Cleaning Solutions based on Hydrogen Peroxide for use
in Silicon Semiconductor Technology”, RCA Review, 187-206, June 1970

2 K. E. Bean, “Anisotropic Etching of Silicon”, IEEE Trans. Electron Devices, ED-25,
1185-1193, 1978.

3 Stephen A. Campbell, The Science and Engineering of Microelectronic Fabficfln,
Oxford University Press, 1996

4 M. Ulczynski, Low-Temperature Deposition of Transparent Diamond Films with a
Microwave Cavity Plasma Reactor, PhD Dissertation, Michigan State University, (1998).

5 K. P. Kuo and J. Asmussen, “An Experimental Study of High Pressure Synthesis of
Diamond Films using a Microwave Plasma Reactor”, Diamond Relat. Mater. 6, 1097-
1105, 1997.

6 H. Windischmann, G. F. Epps, Y. Cong, and R. W. Collins, “Intrinsic Stress in
Diamond Films Prepared by Microwave Plasma CVD”, J. Appl. Phys., February 1991 PP
223 1-2237.

7 D. L. Kendall, “Vertical Etching of Silicon at Very High Aspect Ratios”, Ann. Rev.
Mater. Sci., Vol. 9, 373-403, 1979.

8 K. E. Bean and W. R. Runyan, J. Electrochem. Soc., Vol. 124 50-12c, 1977.

9 M. Josse, D. L. Kendall, “Rectangular-profile diffraction grating from single-crystal
silicon “, Applied Optics, Vol. 19, Issue 1, p.72, January 1980.

72

Chapter 4: Modeling Approach

4.1 Introduction

This chapter describes the modeling of the transmission properties of the Fabry-Perot
device to extract physical parameters from measurements, to predict the performance of
the device, and to guide the actual fabrication of the device. The chapter details the
modeling used in this thesis, on both a mathematical level, as well as discussing the

MATLAB code used to implement the mathematical model devised.

4.2 Motivation for Developing a Different Model

The goal of the model developed for this research is to incorporate the effect of surface
roughness into the matrix model for one-dimensional transmission through multi-layered
optical structures. The treatment of surface roughness in this research neglects near field
effects and is based on a superposition of plane waves. The generally good match
between the model presented here and the measurements taken for this research shows
that this approach can lead to meaningful results. Chapter 2 showed two different matrix
models for calculating transmission through a multilayer structure. Either approach can
be modified to include surface roughness effects in the simulation, although this chapter
will rely primarily upon the transfer matrix technique for the mathematical development

of the model.

73

Before showing the model developed for this research for multilayer structures, it is of
interest to note that a result in the literature is based on an incorrect use of Filinski’s
results for a single interface. The development of this model illustrates this point, as well
as setting the stage for development of the model devised for this research. First,
consider the case of a smooth slab with no surface roughness. Figure 4.1 shows a smooth
slab with thickness d and index of refraction n1 separating two serni-infinite materials

with indices of refraction no and n2.

 

 

 

 

 

A
130+ 130' no
1
131+ E]. E d n]
1
132+ ’12

 

Figure 4.1 a smooth slab

The incident electric field is shown as Eo+, and the reflected field from the slab is Ed.
The transmitted field is shown as E2’. Using the transfer matrix approachl'2’3, detailed in

Chapter 2, for the case of smooth surfaces, the relationship between Eo+, Ed, and E2+ is

E3 :5 E; [4.1]
E5 0

74

found from equation [2.83] to be:

Where the transfer matrix S is found by:

l — r WI 1 _
’01 ’01 ’01110 " ’01’10 0 e ‘12 ’12 ’12’21 — ’12’21

 

with
27mld
= 4.3]
<4 , 1
Using equations [4.1]-[4.3], the field transmission can be found as:
E+ t t
2 = . 01 ‘2 . [4.4]

 

 

E0 61¢] ”12003—1“
Which leads to the well known equation for power transmission through a single thin

layer:

T: 2 00122012)2 [45]
1+(’12) (’10) "2’12’10005(2¢1)

45,67

In the literature , equations [4.4] and [4.5] have been modified by multiplying the

electric field reflection term, no, and transmission term, to], by the square root of the
Filinski correction factors, equations [2.98] and [2.106], in order to simulate transmission

through a thin slab with surface roughness on one side:

_(2w(n0-n1)]2
40
t t e
TR = 12 01 2 [4.6]
4am] _l[4”a’l

1+r122r1ire[ 1° i ‘2’12’106 2 2° ] 008(28)

 

Alternatively, the square root of the Filinski exponential terms have also been used to

modify the Fresnel coefficients in equation [4.2] for the first interface”. This has also

75

been suggested as a general method of including the effects of interface roughness into

the problem of multilayer optics, and also leads to equation [4.6].

However, equation [4.6] does not accurately account for the phase of the wave inside the
film. The Filinski correction terms are based on the integration of equation [2.103] being
carried out over two semi-infinite media with a single interface. The situation is different
for a thin film between two serrri-infinite media because of the effect of the second
interface. For this case, as well as the case of multiple layers, a numerical integration

approach must be used, described as follows.

4.3 Model Developed for this Research

 

 

 

 

 

A B "0
1 f
: dam) 5
E h i
Ly t¥——— 5.1..
i d(B) j dmean j
E i n 1 i
v v v
"0

 

 

Figure 4.2 Two paths through a rough thin film

76

Figure 4.2 shows two paths through a thin film with surface roughness on one side.
Normal incidence is assumed. The roughness of the layer is assumed to be Gaussian in
nature with zero correlation length, and with a mean value of the film thickness dmean.
The length of the optical path through the film is denoted by d, and since the two paths
shown in the figure are of different lengths, they are denoted as d(A) and d(B). A plane in
space above the rough surface is chosen as a reference, located d",- above the second,
smooth surface of the thin layer. The distance from this plane to the rough surface, do, is
also a function of the path length through the layer. Thus the two lengths are labeled

do(A) and do(B). For any path through the slab,
d (path n)+ d0 (path n) = constant = d ,4 [4.7]

where dnfis independent of path through the sample. The transfer matrix technique can
now be applied to this system, and the transmission through (or reflection from) the slab

can be calculated.

In Figure 4.2, a phantom layer of thickness do is essentially created to account for the
phase change in the wave as it travels from the reference plane to the actual interface.
Since there is not an actual interface at the reference plane, the transmission is equal to
unity, and the reflection is zero. However, the wave travels through a distance do
resulting in a phase change, (to, of:

2 d
¢o = 10%! [4.8]

Thus, the transfer matrix of the system, S, can be found by:

77

S _(1)[1 0][ei10 0 ][ 1 M: 1 —’10 ][ei¢l 0 J.”
— 1 0 1 0 ”lb t r t t -r r 0 ‘WI
6 01 01 0110 0110 e [4.9]
...[ 1 I 1 "21 ]
’12 ’12 ’12’21 — ’12’21

The transfer matrix can be simplified to:

 

5 = [4.10]

* =l<

1 ei‘yoew)1 — rlorlze'.¢0e_i‘zil *
101112
Where only the (1,1) element of S is shown, since it alone determines transmission
through the layer. The fraction of the field transmitted is related to the inverse of this

element as:

+
E2 ‘01’12

—:t:

E; emotei¢l

 

. . 4.11
" ’10’1231‘1’0‘3—w>1 [ 1

Following the procedure which lead to equation [2.103], a mean ratio of E2’/ Eo+ can be

calculated by considering a Gaussian distribution of layer thickness, d. This results in the

expression:

 

 

 

Eo’ -..
42,716.. .. “”1“?” [4.121
= tmtlze l e ‘7 d(d)
om -... 1.2—$6140) -i2’,fl(n0+n1)
‘3 —’10’12e

where 01s the RMS value of the surface roughness. Note that do has vanished,
commensurate with its creation as the thickness of a phantom layer, and that when

equation [4.12] is multiplied by its complex conjugate, the term containing dmf will also

78

vanish. If equation [4.12] is used in equation [2.105] to calculate the transmitted power,

the result is different than equation [4.6].

Figure 4.3 shows the results of using equation [4.12] in equation [2.105] to solve for
transmission, as well as a plot of equation [4.6] for the same system. The system shown
in the figure is a 111m thick slab of diamond with 20nm RMS surface roughness on one

side, with serrri-infinite media on either side of the slab with index of refraction equal to

one.

79

0.95 I l' I I T I I I I

 

 

- -------- - using [4.6]
0-9' —— using [4.12] ,.--. -

 

 

 

0.85 r g "1 .

.0
m
l
1

Transmission

 

 

0.5 1 1 r 4 r r r r r
500 550 600 650 700 750 800 850 900 950 1000
Wavelength (nm)

 

Figure 4.3 Two transmission plots for a rough diamond slab. In both cases, thickness = 111m,
roughness = 20nm. The difference between using the approach of equation [4.6] and that of using
[4.12] is readily apparent.

4.4 Extension to Multiple Layers

The concept shown in the previous section of creating a phantom layer and solving
equations [4.12] and [2.105] can be applied to multilayer systems. To do so, matrices
corresponding to additional layers are inserted in equation [4.9] while retaining the

phantom layer and ordering the matrices appropriately for the physical system being

modeled.

80

Here, an example is shown of how to write the transfer matrix for a system that includes

one rough interface in a multilayer stack.

 

 

 

 

 

 

1, .. r
| 0 t
M E
, id. F’fi m E
1 fkfl ldcon
id [drawn 1
. i . a
1, .. 1
r 3 :
Y V
"0

Figure 4.4 A multilayer structure with a rough interface

Figure 4.4 shows a multilayer structure with roughness at one interface, surrounded by
two semi-infinite media. The layer with surface roughness is represented by index of
refraction n2, and the layer with index of refraction n1 coats the n2 layer conforrnally, so
the roughness is seen at the surface of the structure as well. The layer with index of

refraction n3 is assumed to have smooth interfaces on both sides.

The transfer matrix S for this system is then:

81

s...e""° 0 (2][1 —r.. [at 0

0 6"“) ’01 ’01 101110 " ’01’10 0 6"“)1

1 — “’2 1 _
“ll ,. til [...
K112 ’12 ’12’21‘02’21 0 e ’23 ’32 ’23’32—’23’32

L 0 (“’3 ’30 ’30 ’30’03"30’03

where tip is taken from equation [4.3], $1 , ¢2, and (123 are the phase shifts corresponding to

 

the layers which they represent. The (1,1) element of this matrix can be determined, and
an integral corresponding to equation [4.12] can be generated and solved numerically to
calculate the field transmission. Equation [2.105] can then be employed to find the

transmitted power of the system.

One can also use MacLeod’s technique to calculate transmission through the structure in
Figure 4.4. Under the same assumptions and definitions that lead to equation [4.13], the

transmission using the MacLeod technique would be calculated by using the matrix:

"0

i . i .

[B] _ cos 50 —— srn 50 cos 51 Z— srn 51

— I o. 0
inc sin 50 cos 5, in] sin 51 cos 51

. . [4.14]
c0852 —l—sin 52 cos53 —l—sin53 [ 1 ]
n

i212 srn 52 cos 52 his srn 53 cos 53

where 5, is calculated according to equation [2.44], using the appropriate thickness and
index of refraction for the layer in question. When the matrix in equation [4.14] is

calculated, it can be used to calculate the field transmission with the following equations:

2"0

t=-—-— [4.15]
noB‘l’C

82

The field transmission can then be used in equation [4.12] to find the field transmission
of the system considering surface roughness. Equation [2.105] is then used to find the

transmitted power of the system.

4.5 Applying the Model

Appendix A shows the MATLAB code for a program named modexampm. The goal of
the model is to produce a plot of transmission versus wavelength. This program will be
used to illustrate how the model works and the effects that various parameters have on
the output. This program simulates transmission through a thin slab of material with
roughness on one interface and numerically solves the integral problem for multilayered
structures as setup previously in section 4.3. It uses the MacLeod style matrix method

instead of the transfer matrices used to show the development of the model.

The numerical approximation of the integration gives rise to some parameters which are
not present in the analytical formulation of the problem. Typically, the selection of these
parameters involves a trade-off between accuracy of the simulation and computational

time for the simulation.

4.5.1 Setting Up the Variables

The program opens with the code (comments omitted here):

clear

dnom=l.0*10"-6;

n1=1;
n3=1;

83

The nominal thickness of the diamond layer is given by dnom. The units here are meters.
The indices of refraction for the two semi-infinite media are n1 and n3. Here, they are

chosen to be unity, corresponding to free-space.

4.5.2 The Gaussian Distribution

For this thesis, the surface roughness at the material interface is first modeled as
following a Gaussian distribution. The program can be modified to model the roughness
by any distribution. However, Gaussian distributions are often considered a good first
approximation for many processes if the exact distribution is not known. In fact, as
shown in Chapter 6 with AFM measurements, most of the PECVD films in this study
indeed do exhibit essentially a Gaussian distribution in their surface roughness. The issue

of deviations from a Gaussian distribution is treated later.

The form of the continuous first order Gaussian distribution without higher order terms is
given by equation [2.94] and repeated here:

1 20
c(z)=——e [4.16]

The subscript c is to emphasize that this is for a continuous distribution. It should be
noted that this expression is for 2 having a mean value of zero. If z instead has a mean

value of Zo, the expression changes to:

84

e 20 [4.17]

 

1
0'27:

 

Wc(z):

The distribution w,.(z) can be interpreted in the following manner: the probability of
finding z between 21 and z2 is expressed as:
22
p: [wc(z}1z [4.18]
Z1
The distribution wc(z), as expressed here, is normalized. That is, if it is integrated from
=-oo to co, the result is exactly equal to one. This can be interpreted as meaning that the
probability of w(z) taking on some value between -oo and 00 is 100%. This is important,
because the thickness of the diamond films will be modeled by a Gaussian distribution in
this thesis. At any every position on the film, the thickness must have some value. As
earpression [2.15] stands, it leaves open the unphysical possibilities of a negative film
thickness, as well as nearly infinite film thicknesses. However, the nature of using the
matrix model to solve the transmission through the structure can easily eliminate these
unphysical possibilities, due to the fact that a discrete version of the Gaussian distribution

must be used.

A discrete Gaussian distribution with j possible values of z can be calculated by:

_(Zi-20)2
w'(i=1---j)=e 20 [4.19]

However, this expression is not normalized. It can be easily normalized by calculating

the normalization coefficient cN :

85

j
CN = ZW'li) [4.20]
i=1
Thus the normalized expression for this discrete Gaussian is:

(Zr-20 )2

w(i=1---j)=—e 20 [4.21]
Interpreting the meaning of the discrete Gaussian distribution is different than for the

continuous case. The probability of finding 2:2; is simply w(z).

The modexampm MATLAB program listed in Appendix A carries out the calculation

and normalization of the distribution of thicknesses with the following loop:

for n=1zdpp+l;
d(n)=dstart+((n-l)*incrp);
terml=((d(n)—dnom)“2)/(2*(sigma22));
dt(n)=exp((-l)*terml);
nm=nm+dt(n);

enad

dt=dt. /nm;

The loop contains some parameters which thus far have not been mentioned.

The number of terms calculated in the distribution is determined by dpp. For this work, a
tYpical range of values for dpp is 20 to 40. Figure 4.5 shows the error between
Subsequent simulations as a function of dpp for typical values of surface roughness
Observed in this work. A point on this curve is calculated by, for example, calculating the

no point transmission curve for dpp=3 and again for dpp=4. The value of the error at

86

dpp=3 is then found by summing the squares of the differences between the two
transmission curves. Mathematically, this can be expressed as:
"0
E rror(dpp) = Z [pop+1(n)- pop (71)]2 [4.22]
n=l
Thus, the smaller the error becomes, the less impact additional terms in the Gaussian

distribution contribute to the calculated transmission.

 

 

 

 

 

‘1- T l l j l I I I I I
< + 10nm RMS
100 1 -e—15nmRMS _
-e.- 25nm RMS
.2
10 ~ -
§ 10'“ . _
0‘51
10" - -
10" - -

 

 

 

24581012141510202224
dpp

Figure 4.5 Simulation Error as a function of dpp.

As can be seen from Figure 4.5, for each value of surface roughness, the curve changes
Slope at some point, indicating a regime of decreasing return in accuracy as dpp

increases. Thus for this work, all simulations use dpp values of at least 20. Simulation

87

time increases with dpp, so a plot such as Figure 4.5 can be useful in minimizing

simulation time.

Returning to the parameters in the MATLAB code, the nominal value of the film

thickness is given by dnom.

The parameters dstart and incrp require slightly more explanation. Since there are only a
finite number of points in the discrete distribution, it will span only a certain range of
layer thicknesses. In the program, this range is the distance from dstart to dstop, which
are calculated based upon dnom and sigma. For this research, the range spanned by the
distribution is chosen to be a simple function of the RMS surface roughness, 0'. The
starting point of the distribution, and the increment between points are calculated by the

following lines of code:

dstart=dnom-(4*sigma);
dstop=dnom+(4*sigma);
dpp=10;
incrp=(dstop-dstart)/dpp;

As mentioned previously, the nature of having non-infinite start and end points to the
distribution adds physical reality to the model based on the film characteristics. The units

of dstart, dstop, dnom, incrp, and sigma are all meters.

The use of the distribution that the program calculates will be discussed in more detail

later, but for now, it may add clarity to say that the distribution of layer thicknesses will

88

be calculated, then the transrrrission through each thickness will be calculated, and finally
the transrrrissions summed in a weighted manner with respect to the distribution. This

weighted summation is a numerical approximation to the integral in equation [2.12].

4.5.3 Calculating the Transmission

The transmission is calculated with a loop on a variable representing the wavelength.

There is first a small set of code to set up some variables for the loop.

numpoints=400;

1amstart=400*10“-9;
lamstop=l600*lO“-9;

incr=(lamstop—lamstart)/numpoints;

The variable numpoints contains the number of data points that will be calculated along
the wavelength axis. This number should be selected to generate a smoothly varying
transmission vs. wavelength plot. For a given film thickness, the wider the range of
wavelengths considered the more fringes will appear in the transmission. Also, thicker
films will have more fringes in a given range of wavelengths, and thus it is appropriate to
scale numpoints approximately with film thickness and range of wavelength. For the
range of film thicknesses and wavelengths considered in this thesis, a typical value of
numpoints that yields a smooth looking transmission curve is from 300 to 400. More

points can be calculated, but the simulation time increases accordingly.

89

The variables lamstart, lamstop, and incr are analogous to the variables dstart, dstop, and
incrp as explained in the previous section. However, these variables are for keeping track
of the wavelength, 1, instead of film thickness. The variables lamstart and lamstop
should be chosen to cover the range of wavelengths of interest. The units of lamstart,

lamstop, and incr are in meters.

Thus, for the example in Appendix A, 400 transmission data points will be calculated,

with the first being at 400nm, the second at 403nm, the third at 406nm, and so on.

After establishing the wavelengths for which the transmission will be calculated, the first

loop begins with the following code:

for n=lznumpoints+l;

lam(n)=lamstart+(n-l)*incr;

The variable n is an integer which maps to wavelength in conjunction with the vector
lam. The n‘” element of the vector lam is calculated by the second line of code shown
above. After the wavelength is calculated, the index of refraction for the diamond film,
722, is calculated for that particular wavelength by solving the Sellemeier equation. This

part of the code is in Appendix A, but not shown here.

The next step is to perform a numerical approximation to the integral in equation [2.12].

As has been previously stated this is done with a nested loop which carries out a weighted

summation. The code which accomplishes this is listed below:

90

Tplnl=0;

for j=l:dpp;

phiair(j)=(2*pi*nl*(d(dppl—d(j)))/lam(n);
air=[cos(phiair(j)) (i*sin(phiair(j))l/nl;

i*n1*sin(phiair(j)) cos(phiair(j))];

phid(j)=(2*pi*n2(n)*d(j))/lam(n);
D=lcos(phid(j)) (i*Sin(phidlj)))/n2(n);
i*n2(n)*sin(phid(j)) COS(phid(j))l;

BC=air*D*[1;n3l;
B=BC(1);

C=BC(2);
t=(2*nl)/((n1*B)+C);
Tplnl=Tplnl+ldt(j)*t);

Trlnl=Tplnl*conlep(n)):

Again, this loop is nested inside the first 100p on the variable n. This nested 100p

operates on the variable j, which is tied back into the Gaussian distribution calculated

earlier. The variable j is tied to the thickness of the layer being considered, as well as the

probability of the film having that particular thickness.

The variables phiair and phid represent the phase shift experienced by the wave as it

travels through the appropriate layer. The thickness of the diamond layer is calculated

earlier with the Gaussian distribution, and is denoted by (10). The distance that the wave

travels through air before reaching the diamond is the maximum thickness of the

diamond film, d(dpp), minus the particular thickness of the film being considered, d(j).

In these calculations, i is the imaginary unit.

91

These phase shifts are needed in the calculation of the matrices air and D. These

matrices are calculated by equation [2.53].

Next, the vector BC is calculated, using MacLeod’s nomenclature. The field
transmission of the wave, t, is then calculated using B and C. Here, the calculation of t
takes the square root of equation [2.69], using the fact that n, and n3 are both equal to

unity.

The field transmission is then multiplied by the probability of the layer being d(i) thick,
dt(]) and then summed with Tp(n). Tp(n) simply serves as a place to put the sum of the

t*dt(i) product while the loop is running.

When the loop on j has finished and the total transmitted field is accounted for, the
transmitted power for the rough layer, Tr(n), is calculated. This is equivalent to solving

equation [2.105] for the transmitted power.

The program then continues the loop on n, until the transmitted power has been
calculated for the entire range of wavelengths selected prior to starting the loop. At the
completion of the loop, the transmission curve is stored in two vectors. The fraction
transmission is stored in the vector Tr, while the corresponding wavelengths are stored in

the vector lam.

92

4.6 Summary

This chapter began by showing that a result in the literature has an inaccuracy in its
derivation. Then, a model was developed which eliminates the inaccuracy of the
previous model. Finally, key points of a numerical solution devised to implement this
model were discussed. The program presented in this chapter uses the MacLeod matrix
approach, however, the transfer matrix approach may also be used with equivalent
results. Appendix A lists the full text of this program, as well as a program which solves

the problem using the transfer matrix technique.

93

References:

1 0.8. Heavens, Optical properties of Thin Solid Films Dover, New York, 1965.

2 C. L. Mitsas and D. I. Siapkas, “Generalized matrix method for analysis of coherent and
incoherent reflectance and transmittance of multiplayer structures with rough surfaces,
interfaces, and finite substrates”, Appl. Opt. 34, 1678-1683, 1995.

3 Charalambos C. Katsidis, Dimitrios I. Siapkas, “General Transfer-Matrix Method for
Optical Multilayer Systems with Coherent, Partially Coherent, and Incoherent
Interference” Applied Optics-OT, Volume 41, Issue 19, 3978-3987, July 2002.

4 See Chapter 2 Ref. 25 (Gatesman).

5 See Chapter 2 Ref. 26 (Robbins).

6 See Chatper 2 Ref. 27 (Ying).

7 M.J. Ulczynski, B. Wright, D.K. Reinhard, “Diamond-coated glass substrates”,
Diamond and Related Materials 7, 1639-1646, 1998.

8 W. H. Southwell, “Modeling of Optical Thin Films”, Vacuum & thinfilm, May 1999.

94

Chapter 5: Experimental Results and Comparison with Theory Part 1

5.1 Introduction

In this chapter, the samples produced for this research are briefly discussed, and the
optical measurements made on the samples are presented. Before the measurements are
presented, the values of the optical constants for gold and other materials used in this
work are discussed. Then, the optical transmission measurements made on the samples,
as described in Chapter 3, are compared with the theory developed in Chapter 4. After
the measurements on the samples have been presented along with their relevant
simulations, further simulation is explored to investigate the affects of certain physical
device characteristics on the optical performance of the device. All of the simulations
presented in this chapter assume a Gaussian distribution for calculations involving

surface roughness. The treatment of non-Gaussian distributions is considered in Chapter

6.

5.2 Samples

Appendix B has a detailed list of most of the samples made in the course of this research.
A few samples have been omitted since they concerned depositions on small pieces of
wafers to test seeding techniques in a non-quantitative fashion. Fabry-Perot structures
were not fabricated on all of the samples in this Appendix. Before the work on the

Fabry-Perot wafers began, it was necessary to determine, and practice, seeding

95

techniques and deposition parameters to yield films of high optical quality. Additionally,
work on various sizes of substrates was explored, along with some exploratory research
in using different quartz ring heights. Eventually, the technique described in Chapter 3
was determined to be the best for producing films for the Fabry-Perot application as

examined in this thesis.

Several challenges were overcome in developing the fabrication technique discussed in
Chapter 3. Chapter 3 discusses the thermal stress in the diamond films and the solution
obtained for that problem. Before the increased diamond deposition temperature was
established, an alternate method was explored. This method was to deposit the diamond
on an oxide layer. The thought was that since the thermally grown oxide layer was under
tensile stress, and the diamond layer was under compressive stress, the two stresses may
counteract each other enough to result in flat windows when the silicon was through-
etched. This method was implemented on wafer RB2K-2 with 0.48pm of diamond over
1.4]1m of 3102. However, only 29 of 138 windows had an intact film and these were
highly wrinkled. Next, the deposition temperature was increased in order to reduce the
diamond compressive stress as described in Chapter 3. The higher deposition
temperature provided a solution to the wrinkling films, so no further work in the growth

of diamond on an oxide layer was performed.

It is interesting to observe a plot of deposition temperature as a function of deposition

pressure. Figure 5.1 shows such a plot. The data points in this plot were taken from the

“FB” samples listed in Appendix B.

96

 

 

660 . O o 2" wafer temps -
X X 3" wafer temps ..---"

 

 

 

e
u
e" ‘
.....
......
.....
.....
.........
'-
’-
u
a"
0..
a"
...........
.-
.....
e
0..
o"
a"
o'-
.'..
a"
s"
o'-
I".

580-

M“
‘5' "v1

560 ..

 

 

20 25 30 35
Deposition Pressure (Torr)

 

Figure 5.1 Deposition Temperature as a function of Deposition Pressure for MCPR Configuration 7
with a 59mm quartz tube height and approximately 1kW of incident power.

The data in Figure 5.1 was taken using the exact conditions that the films were deposited
under for this research. The deposition system was in MCPR Configuration 7 with a
59mm quartz ring height as describer in Chapter 3. Approximately 1kW of power was
incident upon the system. The probe length and sliding short were adjusted for minimum
reflected power; typical values for these parameters are given in Chapter 3. Two series of
data points are shown in the plot, one on 2” substrates and one on 3” substrates.
Additionally, straight lines fit to the data are shown in the plot. As can be seen in the

Figure, depositions on 2” substrates tend to have slightly higher measured temperatures

97

than those on 3” substrates. Also, for the temperature regime considered here, the
temperature of the deposition tends to follow an approximately linear relationship with

deposition pressure, for each substrate size studied.

A problem on some early samples was the formation of small holes in the diamond films
during the KOH etching. This would sometimes lead to relatively large-area failures in
the film, since the KOH was able to get beneath the diamond and etch the silicon from
both sides of the wafer. The original seeding technique for this work was less elaborate
than described in Chapter 3. It consisted of simply removing the wafer from the package,
scratch seeding it with diamond power, and wiping the wafer clean with a KimWipe.

The more thorough seeding steps discussed in Chapter 3 were implemented along with
higher deposition temperatures, and the combination of these two techniques greatly

reduced the appearance of the small holes in the diamond film during KOH etching.

This chapter looks primarily at Fabry-Perot structures from two wafers in particular,
RB2K8 and RB2K9. These two wafers yielded the diamond films with the best optical
properties of those deposited for this work. These two wafers were intended to be
identical to each other, although as seen in Appendix B, RB2K9 ended up with a thinner
film than RB2K8. As is discussed later in this Chapter, no further wafers were fully

fabricated for Fabry-Perot measurements after RB2K9.

98

5.3 Optical Constants

For this research, the index of refraction for diamond is always taken from the Sellemeier

equation as given by [2.11], and the index of refraction for air is assumed to be unity.

Variations exist in the published optical constants for gold1’2‘3. In this work, two sets of
optical constants are considered for gold. Palik3 gives the constants most often cited in
the literature. However, models using the Kingslake2 data often fit the measurements

made for this research better than those using the Palik data.

Possible explanations for the variations found in the published data include sample
preparation, measurement technique, purity of the gold, and assumptions made in
calculations of n and k from the measurements made on the samples. It stands to reason
that perhaps the data compiled in Kingslake was based on experimental conditions and

assumptions closer to the work in this thesis than Palik’s data was.

Figure 5.2 shows a plot of the Kingslake and Palik data for the index of refraction, n, as a

function of wavelength.

99

 

 

 

 

 

 

 

2 r I I I I F I I
o o Kingslake Data
1.8 - —— Palik Data -
1.6 - -
1.4 - -
S
g 1.2 - -
s
a: 1 - -
"5
g 0.8 - -
E.
0.6 - .
O
0.4 - -
0.2 - -
o 0 O O
0 r 1 r 1 1 r r

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Wavelength (micrometers)

Figure 5.2 Data for the Index of Refraction for gold as a function of wavelength.

Figure 5.3 shows a plot of the Kingslake and Palik data for the absorption constant, k,

also as a function of wavelength.
In both cases, the two sets of data are relatively close and follow similar trends. Palik’s

data covers a wider range of wavelengths than Kingslake’s data, but Kingslake’s data

spans a sufficient range of wavelengths for purposes of this research.

100

 

12 T I I I T I I I

 

o o Kingslake Data 0
— Palik Data

 

 

10-

 

Absorption Constant (1/cm)
O)

 

 

 

l l l l l

o 1 1 1
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Wavelength (micrometers)

 

Figure 5.3 Data for the Absorption Constant of gold as a function of wavelength

Notice that in Figure 5.3, the absorption of gold increases with wavelength. This
behavior is not desirable for operation in the IR. However, the deposition of gold films
was readily available at the time of this research, and the films are very thin so an
appreciable amount of light can still be transmitted through the film, even at the longest

wavelengths studied in this work.

5.4 Measurements on Diamond Membranes without Gold Coatings

This section presents transmission measurements made on several diamond windows

from two wafers, RB2K9 and RB2K8. The transmission data in this section was

101

collected before any gold coating was applied to the wafer. Fitting simulation to this data
allows the extraction of the film thickness and the surface roughness. Observation shows

that the fit is sensitive to around lnm in thickness, and 0.5nm in RMS surface roughness.

Figure 5.4 shows the transmission through a diamond window. This window will be
referred to as diamond window 1, or DWI. No gold coatings were applied at this time.

This data was taken on wafer RB2K9 with a Perkin Elmer Lambda 9000 UV-Vis system.

 

=1oo%)

Transmission (1

 

 

 

— measured
simulation ‘

 

 

 

 

L l

1000 1500 21130 2500
Wavelength (nm)

 

Figure 5.4 Transmission versus wavelength for diamond window 1 on wafer RB2K9. The
wavelength axis spans from the UV (200nm) into the IR (2500nm).

Figure 5.4 shows good agreement between the model and experimental measurement,

except for the visible region where the measured minima are higher than expected.

102

Alignment of the sample and mask on the Perkin-Elmer system is difficult as no fixture
exists which is specific to the samples for this research. For larger area samples where
beam alignment is not critical, this discrepancy was not observed. Fitting the simulation

to the data indicates that this window is 695nm thick, and has a surface roughness of

approximately 19nm RMS.

For a more detailed study of the portion of the spectrum where the Fabry-Perot devices
are to be investigated, a different set up was used which covers a smaller spectral range

but is more amenable to aligning the optical beam with the small mask opening.

Figure 5.5 shows transmission through the same window, DWI, but taken with the
Bausch & Lomb equipment as described in Chapter 3. Note that this figure is different
than Figure 5.3 in that the data points are shown as circles here, and the simulation is a

solid line. This data has been corrected for an offset in the measurement.

103

 

 

 

 

 

0'9 _ — calculated >
0 measured
0.85 ~ 0’ _
0.8 ~ -
gs <2)
8
‘.T 0.75 _
E o
g, 0.7 - -
E
U)
C
E 0.65 - -
I'—
0.6 - -
0.55 - -

 

 

 

 

700 800 90] 10]] 1100 12m 1300 1400 1500 1600
Wavelengthmm)

Figure 5.5 Transmission for the same window as shown in Figure 5.4, diamond window 1, but taken
with Bausch & Lomb equipment.

104

Figure 5.6 shows transmission through a diamond window on wafer RB2K8, DW2, taken
with the Bausch & Lomb equipment. This data has been corrected for an offset in the

measurement.

 

 

 

 

 

 

 

 

— calculated J
0.85 - 0 measured
0.8 - -
S?
8 0.75 - .
31
,5 0.7 -
U)
.9
S <>
E 0.55 — -
*—
0.6 - .
0.55 '— 0 d
1100 1200 1300 1400 1500 1500

Wavelength (nm)

Figure 5.6 Transmission versus wavelength for diamond window 2, on wafer RB2K8.

Fitting the simulation to Figure 5.6 indicates that this film is 1.51 pm thick, and has a

Surface roughness of approximately 27nm RMS.

105

Figure 5.7 shows another diamond window on wafer RB2K8, DW3, with data also taken

on the Bausch & Lomb system. This data has also been corrected for an offset in the

 

 

 

 

 

 

 

 

measurement.
09 I I I I I n -
— calculated . .'
0 measured ' .
0.85- ., ,, 0
0
o O
h 013- 0 a 0 «
a; ., “
D
W 0.75- .. 0 4
:3 0
E, 0 J
“-7 .
E
2 "
L“ 0.85- -
.—
0 o
0.8” . o 0 .1
u 0 (’
055- . ’ ,. o .
O 1 t". 1 roov" 1
1100 1200 1300 1400 1500 1500

Wavelength (nm)

Figure 5.7 Transmission versus wavelength for diamond window 3, on wafer RB2K8.

Fitting the simulation to this data shows that the window is 1.641tm thick, and has an

RMS surface roughness of 26nm.

106

Figure 5.8 shows the transmission through a window on RB2K9, DW4, taken with the
Bausch & Lomb system. The measured transmission was shifted down by approximately

2% to fit the simulation.

 

0.95 I I T fir I T

— calculated

0.9a 0 measured 0

 

 

 

 

0.85 [-

0

100%)
O
CD
0

0.75

 

0.7 - .. 0 -

Transmission (1

0.55 - 0 -

o 0

06' 0 3 .

0 .

0.55 . .' 0 Q -t
(I J: .90 1 .9

800 900 1000 1100 1200 1300 1400 1500 1500
Wavelength (nm)

 

 

 

Figure 5.8 Transmission versus wavelength for diamond window 4, on wafer RBZK9.

Fitting the simulation to this data indicates that the film is 938nm thick, and has an RMS

Surface roughness of about l6nm.

107

Figure 5.9 shows another window on wafer RB2K9, DWS, taken with the Bausch &

Lomb system. This data has had no correction applied.

 

0.9 -

0.85 ~

0.8 -

103%)

0.75 - “

.0
N
I

Transmission (1

(b

 

I I I I I I 1
— calculated _
0 measured

 

 

 

 

0

Q)

1’

0 o
w "
0 0 ‘. 0

11 0' e o

 

 

1 1 1 1 1 1 .0 J
930 1000 11m 1200 131]] 1400 15m 1500

Wavelength (nm)

Figure 5.9 Transmission versus wavelength for diamond window 5, on wafer RBZK9.

Fitting the simulation to the data indicated that this film is 1.061tm thick, and has a

Surface roughness of 18nm RMS.

108

Figure 5.10 shows data for another window on sample RB2K9, DW6, taken with the

Bausch & Lomb system. This data has not been corrected.

 

A
I I I I I I I I V

— calculated ' 0
0 measured "

0.9 "' 'l o D

(a. 0

0.85 - .

 

 

 

 

0.8 ~ ’ .1 -

=1co%)
O

0.75

0.7 '- 1) t) 0 q. ..
>

Transmission (1

0.5 o ..

0 0 0

0.55 r." ,, ‘ . o -
O 00

l l

l l J l l l
:10 900 1000 1100 1200 1300 1400 1500 1600
Wavelength (nm)

 

 

 

Figure 5.10 Transmission versus wavelength for diamond window 6, on wafer RBZK9.

Fitting the simulation to this window indicates a film thickness of 977nm, and an RMS

Surface roughness of 18nm.

Some conclusions from these measurements are that RBZKS appears to be a rougher film
than RB2K9, in that measurements on it showed 19nm to more than 25nm RMS of

SUrface roughness, while RB2K9 typically showed less than 20nm. Also, RB2K8

appears to be thicker than RB2K9, with both of its windows showing around l.6p.m of

109

thickness, while RB2K9 showed no windows thicker than 1.06um. A higher surface

roughness on thicker films is consistent with columnar growth of the diamond film.

5.5 Measurements on Fabry-Perot Cavities

Many windows were measured, coated with gold to form resonators, and measured again
for this research. In fact, it took a large amount of trial and error to arrive at the
approximate 5/3 ratio of gold thickness discussed in Chapter 3. Presented in this section

are measurements of two resonators constructed on RB2K9.

Figure 5.11 shows the transmission through the resonator constructed from DWl, the
diamond window shown in Figures 5.4 and 5.5. No correction has been applied to this

measurement.

110

 

I I I I l I I I

 

— calculated
0.08 o 0 measured '

 

 

 

0.07

100%)
p
8

o
o
01
O

 

P

O

9.
U

Transmission (1
.0
O
(O
I

 

 

 

0000000
800 900 1000 1100 1200 1300 1400 1500 1600
Wavelength (nm)

 

Figure 5.11 Transmission through Fabry-Perot resonator l, constructed on wafer RBZK9.

The parameters used in this calculation are a nominal diamond film thickness of 695nm,
an RMS surface roughness of 19nm, and gold thickness values of 21.5nm and 35.8nm on
the smooth and rough sides of the film, respectively. The measurement and simulation
do not track as well as they do for uncoated diamond windows, however, there is a close

enough match to draw some conclusions from the simulation.

Of particular interest for this measurement and simulation are the Q values of the

resonant peaks. The measured peak at 920nm of wavelength has a Q value of 11.5. The

111

measured peak at 1220nm has a Q of 12.3. The simulation shows a Q of approximately

1 2.6 for each peak.

Figure 5.12 shows transmission through a Fabry-Perot resonator constructed using DWS,

the window shown in Figure 5.9. No correction has been applied to this measurement.

 

 

 

 

 

 

 

 

0.07 _ r I I I I _
. — simulated
o 0 measured

0.06 - o -
r; 0.05 - ‘ .
o\
,8.
n O
c 0.04 - .
.5 0 .
.8 0
E 0 03 ~ .
to
<5
1.

o 02 - . o 0 fl

O
O ' o o
0.01 ~ 0 -
o O
0000“ o
1 1 1 1 . o C G 1&4")
1100 1200 1300 1400 1500 1600

Wavelength (nm)

Figure 5.12 Transmission through Fabry-Perot resonator 2, on wafer RBZK9.

The parameters used in this simulation are a diamond window thickness of 1.06m, a
surface roughness of 18nm RMS, and gold thickness values of 23nm and 27nm. This

window shows a somewhat better fit than is seen in Figure 5.11.

112

For the measurements in Figure 5.12, the peak occurring at 1080nm has a Q of 18, and
the peak at 1370nm has a Q of 17. The simulated peaks show Q values of approximately

20 and 20.4, respectively.

The Q values attained in these measurements are not particularly high, and the total
transmission through the cavity is in the neighborhood of 10% or less. Rather than
repeating the fabrication sequence to make more cavities with similar performance, it is

pertinent to investigate what is limiting the performance of the device.

5.6 Using the Simulation to Gain Insight into the Device

While the simulated Q values are somewhat high compared to measurement, they are
close enough that the simulation can be used to identify trends in Q as a function of
physical resonator parameters. The surface roughness of the diamond film is one of the
main limiting factors in attaining good performance from the device. Figure 5.13 shows

a simulation of Q as a function of surface roughness.

113

 

110< I I I I I I 1 I I

100

I
l

80» -

I
1

70

60

I
l

O (unitless)

50

I
l

'L-

40

I
I

30~ a

I
l

20

 

 

l L I l I I

B 8 10 12 14 16 18 20
Surface Roughness (nm RMS)

 

10
0

M.—
A

Figure 5.13 Q as a function of surface roughness for a resonator based on a lnm thick diamond film.

The simulation for Figure 5.13 is based upon a diamond thickness of 111m, and a gold
thickness of 31.5nm on each side of the diamond. The peak studied for the simulation is
at 1265nm. From this simulation, it can be seen that the surface roughness of the
diamond film is critical to the performance of the resonator. Even at a roughness of
around Snm RMS, Q is reduced from that of a perfectly smooth film by a factor of about
2 for the resonator in question. For surface roughness in the range of 15-30nm, like the
films in this study typically show, Q is reduced by a factor of 4 to 5 from the case of

perfectly smooth diamond.

114

A second parameter of concern is the thickness of the gold layers deposited on the
diamond window. Since gold has strong absorption at optical wavelengths, this serves to
attenuate the power of the transmitted beam. However, performance of the resonator
relies on having a reflective surface on either side of the diamond film, so there is a trade-
off between signal attenuation and resonator performance, expressed here as Q. To
investigate this, a new figure of merit can be established. Figure 5.14 shows a plot of Q

multiplied by the peak transmission, QT, as a function of gold thickness.

 

35_ I I ‘Vw-° I I I I I

0

I

25 "' 0 .,

OT (unitless)

15 r o '

I
6
l

10

 

 

l l l

0 10 20 30 40 50 60 70 Ill 90
Gold Thickness (nm)

Figure 5.14 QT as a function of gold thickness for a resonator based on a 111m thick diamond film.

The simulation in Figure 5.14 is based on a smooth diamond film lum thick. The gold
thickness shown on the x-axis is the thickness of the gold layer on each side of the

diamond film. The wavelength of the peak studied in this simulation actually shifts with

115

increasing gold thickness, however, it is in the range of 1 to l.2p.m. In this case, QT
peaks at around 30nm of gold thickness. So having 30mm of gold thickness on each side

of the wafer would represent the optimum design for maximizing QT.

The next simulation, shown in Figure 5.15, also creates a plot of QT as a function of gold

thickness, in this case, however, 18nm of RMS surface roughness is added to the

simulation.

 

QT (unitless)

 

 

l 1

5 10 15 20 25 30
Gold Thickness (nm)

 

 

Figure 5.15 QT as a function of gold thickness for a resonator based on a 111m thick diamond film
with a surface roughness of 18nm RMS.

The resonator studied in this figure is based on a 111m thick diamond window with 18nm

of RMS surface roughness. As can be seen in this figure, the addition of surface

116

roughness changes the gold thickness that maximizes the figure of merit QT. In this case,
the simulation indicates that extremely thin gold layers may lead to optimal performance,
however, layers this thin are not known to demonstrate the same optical constants as bulk
gold“, so the simulation may not be accurate in this range of thicknesses. The conclusion

can nonetheless be drawn that thinner gold layers may give better results for rough

diamond membranes.

The final simulation for this chapter, shown in Figure 5.16, is similar to that shown in

Figure 5.13, but in this case, thinner gold layers are used.

 

35
<

30

Q (unitless)
m
U"!

B

15

 

 

 

l 1

2 4 6 8 10 12 14 15 18 20
Surface Roughness (nm RMS)

1 l l

—
—

 

10
0

Figure 5.16 Q as a function of surface roughness for a resonator based on a 111m thick diamond film.

117

The simulation shown in this figure uses a 111m thick diamond window with 15nm thick
gold layers on each side. As can be seen from this figure, the Q is even lower than that
shown in Figure 5.13, however, since the peak transmission was also considered in this
selection of gold thickness, more power should be transmitted through the resonator in
this case. Figures 5.17 and 5.18 compare these two simulations further. Figure 5.17

shows the Q as a function of surface roughness for both models.

 

fir

—e— 16nm Gold Layers
100 - —e.- 31 .5nm Gold Layers

 

110‘ T I I I I I

 

 

l

 

I
1

90

I
l

80

I
l

70

1
1

60

O (unitless)

50- -
40- ~
<
30

20

 

 

 

l 1

2 4 6 8 10 12 14 16 18 20
Surface Roughness (nm RMS)

1

 

10 l 1 L 1
0

Figure 5.17 Q as a function of surface roughness for both gold layer thicknesses considered.

Figure 5.18 shows the peak transmission as a function of surface roughness. This plot
shows that the thinner gold layers should pass a larger percentage of the optical power,

despite having peaks that are less sharp.

118

 

I T r

 

—e— 16nm Gold Layers
—e.— 31.5nm Gold Layers

 

 

 

100%)

.C’
is

l‘\

 

Peak Transmission (1
_o O
m 0.)

1:3
——8
I

 

 

5
O 1 1 1 J l i l l 1
0 2 4 6 8 10 12 14 16 18 20
Surface Roughness (nm RMS)

 

Figure 5.18 Peak transmission vs. surface roughness for the two gold layer thicknesses considered.

What can be seen from Figures 5.16, 5.17, and 5.18 is that achieving a high Q with
realistic, as-grown surface roughness values is not possible. This means that to increase
the Q values of these cavities to reasonable levels, without sacrificing almost all of the
transmitted power, a post-processing technique, for example a mechanical polishing step,
is needed. This is beyond the scope of this research and was not investigated
experimentally. However, using simulations such as those in this section can give an

estimation of what surface roughness must be achieved in order to reach a desired Q

value.

119

Another interesting route for future exploration is to use reflective layers made from
either metals that are less lossy than gold, such as silver, or preferably lossless materials.
The application of lossless multilayers as reflective coatings for Fabry-Perot resonators,

however, is a non-trivial and is beyond the scope of this works.

5.7 Summary

In this chapter, the samples fabricated for this research were briefly discussed, as well as
the optical constants of the materials specific to this work. After this discussion,
measured results were shown with their corresponding simulations, and the parameters
extracted from the simulations were listed. Finally, the simulation was used to show that
the current method of fabricating the resonators will result in relatively low Q values,
although some optimization of the device behavior is possible. The low Q is primarily a
function of surface roughness in the diamond film. The diamond films grown for this
research tend to be relatively smooth for PECVD diamond films, and even if it is possible
to grow the film with somewhat less surface roughness, the simulation shows that
significant gains in Q will not be seen until the RMS surface roughness of the film is in
the range of 5nm or less. This would not be likely in as-grown films using the current
deposition method. This means that to fabricate resonators with significantly higher Q
values, a polishing step is probably needed. Alternatively, one may use a different

growth method, such as that used to grow ultra-nanocrystalline diamond films.

120

References:

1 W. G. Driscoll, W. Vaughan, eds, Handbook of Optics, Sponsored by the Optical
Society of America, McGraw-Hill Book Company, New York, NY, 1978.

2 See Chapter 2 Ref. 34 (Kingslake).

3 ED. Palik, ed, Handbook of @tical Constants of Solids, Academic Press, Inc, New
York, NY, 1985.

4 S. A. Kovalenko, “Dimensional effects in thin gold films”, Semicond. Phys, Quant.
Elect. & Optoelect., V3 N4, 2000.

5 PE. Ciddor, Applied Optics 7, 2328-2329, 1968.

121

Chapter 6: Experimental Results and Comparison with Theory 2

6.1 Introduction

In Chapters 4 and 5, it is explicitly assumed that the samples have a surface roughness
that can be modeled by a employing a Gaussian distribution. An important question to
consider is if this is a good approximation to the reality of the film, since the fit of
Chapter 4’s theory is not perfect with the measurements in Chapter 5. In this chapter,
atomic force microscopy (AFM) measurements are used to show that the surface
roughness displayed by most films is indeed close to a Gaussian distribution.
Additionally, parameters can be extracted from the AFM image. The most interesting
parameter for this research may be the RMS surface roughness, 0'. Higher order

moments of the distribution can also be extracted from the data.

6.2 Samples

AFM measurements were made on three samples, RB2K8, RB2K9 and FB26. The
sample FB26 was not patterned or through-etched like RB2K8 and RB2K9, although the
diamond film was deposited under conditions similar to those studied in this work.

Additionally, some optical transmission data is available for this sample.

As listed in Appendix B, the film on RB2K8 was deposited at 35 Torr for 5.5 hours. The

gas flows were 200 sccm of H2, 8 sccm of C02, and 3 sccm of CH4. A weight gain

122

during diamond deposition of 8.7 mg was recorded for this sample. Optical transmission

measurements indicate a thickness of 1.5um.

Also as listed in Appendix B, the film on RB2K9 was deposited at 35 Torr for 5.5 hours.
The gas flows were 200 sccm of H2, 8 sccm of C02, and 3 sccm of CH4. A weight gain
during diamond deposition of 6.3 mg was recorded for this sample. Optical transmission

measurements indicate a thickness on the order of 0.9um.

The film on the FB26 sample was deposited at the same gas flows, but the pressure was
20 Torr and the deposition was 12 hours in length. At this pressure, the observed
temperature of the substrate during deposition was approximately 590°C. FB26 was
grown as part of a different set of experiments than this research, so no weight gain data
was collected. However, optical transmission measurements show that this film is

approximately 1.4um thick.

6.3 AFM Images

AFM is an interesting instrument, in that it gives a numerical measurement of the sample
topography. This thesis uses these measurements in an attempt to gain insight into the
nature of the diamond film, but the AFM can be used as an imaging tool, as well. Figure

6.1 shows an AFM image of sample RB2K8.

123

 

micrometers

 

micrometers

Figure 6.1 AFM picture of sample RBZK8 (this image will ako be referred to as AFM Image 2)

The horizontal dimensions of the sample are plotted in the X-Y plane, while the Z
dimension is plotted as map in gray scale. The scale bar on the right side of the picture

corresponds to the surface height measurement.

The granularity of the diamond film is clearly visible in this image. The image is
essentially a view straight down onto the diamond film. The AFM image is not optical in
nature, the intensity of the pixels is based on the height of the sample measured at that
point. The image in Figure 6.1 is 5pm on a side in the X-Y plane, and the Z scale

extends from the lowest point measured on the sample up to approximately 200nm above

124

the lowest point. The lowest point on the sample would be black, and the highest point of

the sample would be white, with varying shades of gray in between.

AFM is capable of very high resolution. Figure 6.2 shows an image from the sample

FB26, which is 111m by lum.

0.5

micrometers

0.5

0.7

0.8

0.9

 

0 0.2 7 0.4 7 0.5 0.0 if 1
micrometers

Figure 6.2 AFM imge of the sample F826 (this image will also be referred to as AFM Image 8).

Figure 6.3 shows an image taken at the same location as Figure 6.2, but at a higher

magnification so that each side of the image is 500nm.

125

 

micrometers

 

'0 0.1 7 0.2 0.3 04
micrometers

Figure 6.3 AFM image of sample F326 (this innge will also be referred to as AFM Image 9).

These two figures show interesting images of the grains in the diamond film. By
inspection of the images, the average grain size is clearly sub-micron, in the range of

200-300nm.

Other renderings of AFM images can be generated in addition to top views. Figure 6.4

shows a simulated 3—dimensional view of the same measurement shown in Figure 6.1.

126

e
e
ee
0 e
.0
a
e'.
a
e
e
.e
e
o'-
e
as
.e' e
e
u
e
e
.e

n a n
e e
e e
eeeeee

0.8 . ----- ------- =--

a
e'.
.

 

 

Figure 6.4 Simulated 3-Dimensional view of data in Figure 6.1, sample RBZKS. All axes are in units
of micrometers.

Notice that the aspect ratio of the image in Figure 6.4 is not proper, that is, the Z axis
spans 1pm, while the X and Y axes each span Sum. This leads to the surface appearing

to be rougher than it would if the scales were equal in proportion.

6.4 AFM Data

As discussed in Chapter 3, the AFM used in this research is a Digital Instruments
Nanoscope III. That particular instrument is capable of producing an ASCII file

containing the data from its measurement. The file has a header that contains information

127

on the scaling of the image, the size of the measurement, the number of data points, dates,
miscellaneous settings of the instrument, etc. The actual image is represented by a matrix

of integers. Each element of the matrix corresponds to a pixel in the AFM image.

The physical dimensions of measurements made by the instrument can be determined by
using the data in the file header combined with the matrix of integers. A MATLAB

program was written to do this.

Once the data is in MATLAB, it becomes possible to perform some statistical analysis.
One of the first questions then explored is whether or not the measured surface heights
follow a Gaussian distribution. The next exploration has to do with extraction of

parameters from the measurements, namely the moments of the distributions.

Through a MATLAB program, the distribution measured by the AFM can be plotted.
Figure 6.5 shows a plot of the distribution of surface heights measured for Figure 6.1, as
well as a calculated Gaussian distribution, shown for comparison. There are two
parameters in the calculated Gaussian, the mean value and the standard deviation. In this
case, the mean was chosen to fit the measurement. The mean of the measurement has no
significance to this work. For modeling purposes, the mean is the mean thickness of the
film, which the AFM cannot measure, and must be determined by alternate methods. The
stande deviation was taken from a value calculated by the AFM software. The Y

values are arbitrary.

128

 

 

 

 

1 I I .‘ A I I I I I
3'73;
.0 I ...
0.9 - - c'?‘ . 4
0.8” ... e (e "
l.
0.7 " ,r‘. .1. _
I
0.5 - .~ _
0.5 5 *'
I
0.4 - 1
0.3- ~/
0.2 - . 5'
0.1 -
U I I I

 

1110.

Figure 6.5 Distribution of surface heights for sample RBZKB, as taken from the AFM image shown in
Figure 6.1 (data points), and an ideal Gaussian distribution (black line).

The data for this plot follows something similar to a Gaussian distribution, but that there
is some difference. One interesting observation is that using the measured mean of the

distribution is not the best fit of the pure Gaussian distribution to the measurement.

6.5 Calculations of Statistics from AFM Image Files

Many software packages are available to calculate statistics of a set of numbers, but the
AFM images in this work pose a few obstacles. First, the data is in the form of a matrix

that typically needs to be converted into a single column array, or a double summation

129

could be used to perform the statistical calculations. Secondly, the data should be scaled
properly in order to calculate meaningful statistics. Scaling was addressed previously in
Section 6.4. For this research, the matrix data from the AFM image file was converted

into a single column vector by a MATLAB program.

In this Chapter, four statistics of the distribution are considered. These are the mean
value, the standard deviation, the skewness and the kurtosisl'2'3‘4. The mean value
corresponds to the average film thickness — although the AFM cannot directly measure
this quantity. The standard deviation is a measure of the variation about the mean, i.e. the
surface roughness of the film. The skewness is a measure of how centered, or symmetric,
the bulk of the surface height distribution is about the mean value. The kurtosis is a
measure of how flat or sharp the top of the peak is compared to an ideal Gaussian

distribution.

The vector 2 contains all of the points measured by the AFM in a sequential order. If the

vector has N points, the mean, 2mm, can be calculated by:

zmean : —Z Z1“ [6.1]

The standard deviation of the distribution, 0', can be calculated by:

a- Iiiz-Z [62]
Ivi=1 l

The skewness, skew, of the distribution can be calculated by:

130

skew = "(___—3: (Zi _ zmean) [63]

With this definition of skewness, a pure Gaussian distribution will have a skewness of

zero.

The kurtosis, kurt, of the distribution can be calculated by:

kart = ___—2'2 (21' _ Zmean )4 [6’4]

Using this definition of kurtosis, a pure Gaussian distribution will have a kurtosis of

three. Kurtosis is also sometimes defined by subtracting three from equation [6.4].

A MATLAB program was written that applies the above formulae to the distribution

gathered from the AFM image file, and outputs the calculated values.

6.6 Measurements

The AFM measurements made for this research are now presented. For each

measurement, a top view image is first shown (or in some cases, reference is made to an
earlier figure), and then the measured distribution of surface heights, plotted along with
an ideal Gaussian distribution, which is shown to illustrate any deviation from this ideal

case. Additionally, the mean value of the distribution, standard deviation (which is the

131

RMS surface roughness), skewness and kurtosis for each measured distribution are given.

These statistics are calculated from the actual measurements.

6.6.1 AFM Image 1: RB2K8-1

 

micrometers

 

micrometers

Figure 6.6 AFM Image 1

132

 

 

 

 

 

 

 

 

1 I I I I A I I I I I I
.g .
3...}; ° measured dist. (AFM)
0.9 ~ .«— " a ' —e— Calc. Gaussian dist. —
I. v; 9
0 ' e 3» ii
0.8 - ,a , 0 _
0: ‘11 o
0.7 " . I, r d
0 I: ‘I'.
.t . 0
0.6 "' ...1; 0:. ..
r} T“;-
0 5 ~ " ‘.- 1‘ -
q"w
0.4 - 1,: ~ *‘ -
4'3 ‘4 P
0‘3
0.3 - gr “ -
.4,
0 I? 75 e
't .
0.2 - gr ’ - -
I? ““3.
’l ‘4.-
0-1 ' ...... .31.. '
’ ’J .1533 ém
l‘ .‘-"’ ’__‘d33_”'; i 1 l I I I I INS“ .*l: 1'? c"- v v
1 2 3 4 5 5 7 0 9 10
meters -8
x 10

Figure 6.7 Distribution for AFM Image 1 from RBZK8.

For AFM Image 1, the measured mean is 46.373nm, and the measured standard deviation

is 14.070nm. The measured skewness is 0.4424, and the measured kurtosis is 3.4155.

133

6.6.2 AFM Image 2: RB2K8-2

The image for AFM Image 2 is shown in Figure 6.1. The distribution measured from the

image is shown below.

 

 

 

 

 

 

 

 

L(‘ I L I J

32:. - measured dist. (AFM)
0.9 - °° cf: . -e- Calc. Gaussian dist. .

6;. ‘-;.
0.8 "' ‘o' F e ..
’..‘
0.7 ~ 2 . ° , -
r! °
0.5 - f - 0. . _
lg. E'
0.5 - ._ .
0 . ’
0.4 - ‘ ‘ ‘ -
0 f
0.3 - ' J
0 v
0.2 - 0 -. .
I; . :1. 9
0.1 - 9 g. . .
’I \‘(‘ ‘ .
C 1 1 1 A “’5’5'1-6- s

2 4 5 e 10 ' ' ' ' 12

meters x 10-3

Figure 6.8 Distribution for AFM Image 2 from RB2K8.

For AFM Image 2, the measured mean is 46.668nm, the measured standard deviation is

16.882nm, the measured skewness is 0.5253, and the measured kurtosis is 3.4619.

134

6.6.3 AFM Image 3: RB2K8-3

0.8

 

 

micrometers

   

micrometers

Figure 6.9 AFM Image 3.

135

 

I I l I
‘ . . 0 measured dist. (AFM)
g'g - .vufi’: : —e— Calc. Gaussian dist.

 

 

 

 

 

.3
meters x 10

Figure 6.10 Distribution for AFM Innge 3 from RBZKS.

For AFM Image 3, the measured mean is 48.166nm, the measured standard deviation is

16.727nm, the measured skewness is 0.4238, and the measured kurtosis is 3.1469.

136

6.6.4 AFM Image 4: FB26-l

micrometers

 

micrometers

Figure 6.11 AFM Image 4.

137

 

1 I I I I g ,vI' I I f

0 measured dist. (AFM)
0'9 f —e— Calc. Gaussian dist. ‘

 

 

 

 

 

0.8

I

0.6 -

0.5 -

0.4 -

0.2 h

0.1

 

 

 

2 4 5 a 10 12 14 1515
meters 1110.8

Figure 6.12 Distribution for AFM Image 4 from F326.

For AFM Image 4, the measured mean is 95.870nm, the measured standard deviation is

25.068nm, the measured skewness is -0.0723, and the measured kurtosis is 2.8521.

138

6.6.5 AFM Image 5: FB26-2

micrometers

 

micrometers

Figure 6.13 AFM Image 5.

139

 

1 I I I f *I

0 measured distribution (AFM)
0.9 - -e- Calculated Gaussian dist.

    

 

 

 

0.8 -

0.7 r .7 _

I

0.6

0.5

I
l

0.4

I
l

a:
I
, I
03~ ~ -
. C
l

0.2

I
l

   

 

 

   

I I I I I

1 0.2 0.4 0.5 0.5 1 1.2 1.4 1.5 1.52
meters x104

     

Figure 6.14 Distribution for AFM Image 5 from F326.

For AFM Image 5, the measured mean is 113.22nm, the measured standard deviation is

25.592nm, the measured skewness is —0.0840, and the measured kurtosis is 2.9184.

140

6.6.6 AFM Image 6: FBZ6-3

micrometers

 

micrometers

Figure 6.15 AFM Image 6.

141

 

1 A

I I I I I l I I I I

1 measured distribution (AFM) ,'
-e— Calculated Gaussian dist. . t ‘

 

0.

(D
I

 

 

 

 

 

-7
meters x 10

Figure 6.16 Distribution for AFM Image 6 from F326.

For AFM Image 6, the measured mean is 118.39nm, the measured standard deviation is

26.061nm, the measured skewness is —0.1290, and the measured kurtosis is 2.9908.

142

6.6.7 AFM Image 7: FBZ6-4

 

 

micrometers

 

micrometers

Figure 6.17 AFM Image 7.

143

 

1 I I I I I O I I I I I
1 measured distribution (AF M) g.
-e— Calculated Gaussian dist. :81" ’ _

O

 

0.9

 

     
   

0.8

0.7 -

0.5

0.4 ~

I

0.3

0.2 ~

0.1

 

 

. 3 . _ ' ,. 1 1 1 1 i . ' )
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
meters 11 10-7

Figure 6.18 Dktribution for AFM Innge 7 from F326.

For AFM Image 7, the measured mean is 117.57nm, the measured standard deviation is

28.930nm, the measured skewness is —0.0384, and the measured kurtosis is 2.8918.

144

6.6.8 AFM Image 8: F326-5

AFM Image 8 is shown in Figure 6.2. The distribution calculated from the image is

shown below.

 

 

 

1 l l I o e l I
. measured distribution (AFM) ' : "1
0-9 ' —e— Calculated Gaussian dist. 15‘ ..°‘ . ‘

 

 

 

 

 

meters )1 10'

Figure 6.19 Distribution for AFM Image 8 from F326.
For AFM Image 8, the measured mean is 72.428nm, the measured standard deviation is
24.106nm, the measured skewness is —0.1708, and the measured kurtosis is 2.5308.

Since relatively few grains are shown in AFM Image 8, these statistics probably have

somewhat less meaning than previous images of this sample.

145

6.6.9 AFM Image 9: FB26-6

AFM Image 9 is shown in Figure 6.3. The distribution calculated from this image is

shown below.

 

 

~ measured distribution (AFM) '
0-9 ' —e— Calculated Gaussian dist. ’

 
  

 

0.8 - ° 4
0.7
0.6
0.5
0.4
0.3
0.2

1
I
0
’ ’O.‘ O
0.1- '4’
,9
( f .. ”O
9 I I I I I I I I
9

0 1 2 3 4 5 6 7 8
meters -8

 

 

Figure 6.20 Distribution for AFM Image 9 from F326.

For AFM Image 9, the measured mean is 53.024nm, the measured standard deviation is
21.762nm, the measured skewness is 0.0605, and the measured kurtosis is 2.2014.
Again, since relatively few grains are shown in AFM Image 9, these statistics probably

have somewhat less meaning than previous images of this sample.

146

6.6.10 AFM Image 10: RB2K9-1

 

 

 

micrometers

   

0 1 2 3 4 5
micrometers

Figure 6.21 AFM Image 10.

147

 

0 measured distribution (AFM)
—e— Calculated Gaussian dist.

 

  
   

0.9

I

 

 

 

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

 

 

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.-
meters -?

Figure 6.22 Distribution for AFM Image 10 from R32K9.

For AFM Image 10, the measured mean is 81.598 nm, the measured standard deviation is

29.604 nm, the measured skewness is 0.4948, and the measured kurtosis is 3.5080.

148

6.6.11 AFM Image 11: RB2K9-2

 

 

micrometers

      

0 2 4 6 8 10
micrometers

Figure 6.23 AFM Image 11.

149

 

1 I I I A I I I I

, . 1 measured distribution (AFM)
0.9 ~ 1}“ .1 -e- Calculated Gaussian dist.

 

 

 

1

   

 

 

 

 

0.6 0.8
meters )1 10-7

Figure 6.24 Distribution for AFM Image 11 from R32K9.

For AFM Image 11, the measured mean is 84.198 nm, the measured standard deviation is

26.151 nm, the measured skewness is 0.3172, and the measured kurtosis is 3.0308.

150

6.6.12 AFM Image 12: RB2K9—3

micrometers

 

0.2 0.4 0.5 0.5 1
micrometers

Figure 6.25 AFM Image 12.

151

 

 

' l
o

. I I I I
0. ° measured distribution (AFM)
—e— Calculated Gaussian dist. -

  

0.9

 

  

0.8

0.7

0.5

0.5

0.4

0.3

0.2

0.1

 

 

 

meters -7

Figure 6.26 Distribution for AFM Image 12 from R32K9.
For AFM Image 12, the measured mean is 61.117 nm, the measured standard deviation is
28.944 nm, the measured skewness is 0.9593, and the measured kurtosis is 4.8004. Since

relatively few grains are shown in AFM Image 12, these statistics probably have

somewhat less meaning than previous images of this sample.

152

6.6.13 AFM Image 13: RB2K9-4

micrometers

 

micrometers

Figure 6.27 AFM Image 13.

153

 

 

.. 1.3:. 1 measured distribution (AFM)
0.9 - —e— Calculated Gaussian dist.

    

 

0.0 -

0.5 -

0.5 -

0.3 -

 

 

 

0 0.2 0.4 0.5 0.5 1 1.2 1.4.1.5

Figure 6.28 Distribution for AFM Imge 13 from R32K9.

For AFM Image 13, the measured mean is 81.004 nm, the measured standard deviation is

27.504 nm, the measured skewness is 0.4523, and the measured kurtosis is 3.2733.

154

6.6.14 AFM Image 14: RB2K9-5

micrometers

     

0 1 2 3 4 5
micrometers

Figure 6.29 AFM Image 14.

155

 

 

C 3., 0 measured distribution (AF M)

0.9 -e- Calculated Gaussian dist.

I

 

 

 

 

0.8

0.7

0.5

0.5

0.4

0.3

0.2

0.1

 

 

 

- ' C v v v 0

meters )1 10-7

Figure 6.30 Distribution for AFM Image 14 from R32K9.

For AFM Image 14, the measured mean is 76.164 nm, the measured standard deviation is

26.226 nm, the measured skewness is 0.3731, and the measured kurtosis is 3.0048.

156

6.6.15 AFM Image 15: RB2K9—6

micrometers

 

micrometers

Figure 6.31 AFM Image 15.

157

 

 

2 measured distribution (AFM)
1 —e— Calculated Gaussian dist. -

   
   

0.9

I

 

 

 

0.8

I

0.7

I

0.5

0.4

I

0.3

0.2

0.1 P

 

 

 

0.2 0.4 0.5 0.5 1 1.2 1.4 1.5
meters x104

Figure 6.32 Distribution for AFM Image 15 from R32K9.

For AFM Image 15, the measured mean is 81.842 nm, the measured standard deviation is

24.667 nm, the measured skewness is 0.2825, and the measured kurtosis is 3.0553.

158

6.6.16 Measurement Summary

Nine images have been shown and their distributions analyzed. A Table 6.1 summarizes
the measurements of RB2K8. Recall that RB2K8 has a nominal thickness of 1.5um,

based on optical transmission measurements.

 

 

 

 

 

RB2K8-l RB2K8-2 RB2K8-3 Average
Mean (nm) 46.373 46.668 48.166 47.069
Std. Dev. (nm) 14.070 16.882 16.727 15.893
Skewness 0.4424 0.5253 0.4238 0.4638
Kurtosis 3.4155 3.4619 3.1469 3.3414

 

 

 

 

 

 

 

Table 6.1 Summary of R32K8 AFM measurements.

Table 6.2 summarizes the measurements taken on FB26. FB26 has a nominal thickness
of 1.4um based on optical transmission measurements. Since images 5 and 6 contain

significantly fewer grains than the first four images and their statistics may therefore be

less meaningful, an average of the first four images only is also shown.

159

 

 

 

 

 

 

 

 

 

 

 

 

 

 

33 83 6’3 83 33 6‘3 3 5" 3

N N N N N N a :1; 9;

9‘ 9‘ 9‘ 9‘ 9 9‘ B o 0%

1— N b.) A U1 Ox 0‘8 51 0

O

"h
Mean (nm) 95.870 113.22 1 18.39 117.57 72.428 53.024 95.084 111.263
Std-DCV- (nm) 25.068 25.592 26.061 28.930 24.106 21.762 25.253 26.413
Skewness 00723 -0.084 -0129 00384 01708 0.0605 0.0723 -0.08093
Kurtosis 2.8521 2.9184 2.9908 2.8918 2.5308 2.2014 2.7309 2.9133

 

Table 6.2 Summary of F326 AFM measurements.

Table 6.3 summarizes the measurements taken on RB2K9. RB2K9 has a nominal

thickness of about 0.9um based on optical transmission measurements. Since image 3

from this series contains significantly fewer grains than the other images and its statistics

may therefore be less meaningful, an average of the images not including image 3 is also

shown.

160

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

70 79 79 79 >

0:1 63 00 00 0:1 5 3 E 5

N N N N N N S,“ 7: a

W W :78 79 79 7*: 1» eon

‘P ‘P “.3 ‘9 ‘P ‘P 8 1'» °

.— N w 1:. m G in

a
Mean (nm) 61.761 72.639 43.989 ’ 68.715 63.049 73.349 63.917 67.903
Std. Dev. (nm) 22.407 22.561 20.812 23.332 21.710 22.107 22.155 22.423
Skewness 0.4948 0.3172 0.9593 0.4523 0.3731 0.2825 0.4799 0.3840
Kurtosis 3.5080 3.0308 4.8004 3.2733 3.0048 3.0553 3.4454 3.1744

 

 

Table 6.3 Summary of R32K9 AFM measurements.

It should again be noted that the mean value is listed in these tables only for

completeness, and it does not have any direct significance for purposes of this research.

It is interesting to observe that even though they were deposited under similar conditions,
the films seem to have some noticeable differences in their statistics. First of all, FB26
appears to be much closer to an ideal Gaussian shape, since it’s skewness and kurtosis are
both very small in magnitude, while RB2K8 shows almost five times more skewness and
kurtosis than FB26. RB2K9 shows greater surface roughness than RB2K8, but like

FB26, it shows less skewness and kurtosis than RB2K8.

Also, FB26 is significantly rougher than RB2K8 and RB2K9, having an average standard
deviation of around 26nm compared to RB2K8’s l6nm average and RB2K9’s 22nm

average. It is also worth noting that RB2K8 appears to be slightly rougher when

161

extracting the roughness from optical measurements than when measured via AFM. The
optical fit resulted in a surface roughness range of 19nm - 25nm, whereas the AFM range
was 14 - l7nm. This may be due in part to the fact that the parameter extractions in
Chapter 5 assumed a pure Gaussian and the “extra” roughness compensates for the lack
of higher order moments in the distribution. Also, the optical measurements deal with
two surfaces and other effects, such as inter-grain boundaries and absorption in the
diamond film, which are not included in the AFM measurement. These effects may also

lead to measuring “extra” surface roughness optically.

However, RB2K9 appears to be rougher when measured with the AFM than when
extracting the roughness from optical measurements. This is may be due in part to the
fact that the surface of RB2K9 appeared to have some impurities present at the time of
the AFM measurement. The presence of these impurities is speculative, since the AFM,
as it was used for collecting these images, is purely a topographical tool, and has no
means of chemical or electrical analysis. The surface of the sample was cleaned with
alcohol and a cotton tipped swap, but the apparent presence of foreign objects in the

image remained nonetheless.

Whether or not the objects on the surface of the sample are impurities, it can be seen that

excluding them in the surface roughness calculation leads to values similar to RB2K8.

162

6.7 Including Higher Order Moments in Optical Simulations

It is interesting to consider how the skewness and kurtosis of the distribution may
influence the modeling of optical transmission through the device. Section 6.5 discussed
how the mean, surface roughness, skewness and kurtosis were calculated from the surface
height data measured by the AFM. This section shows how to reconstruct an analytical
distribution given those parameters. The Pearson Type-IV distributions allows one to
specify skewness and kurtosis in addition to the mean and standard deviation and
calculate a distribution based on those parameters. The non-normalized Pearson Type-IV

distribution as a function of x is given by the following set of equations:

 

 

n(x) = exp(A — B tan—1 C ) [6.5]
where A is:
A ..__ 111(b0 +bl(x_xmean)+b2(x-xmean)2) [66]
2192
and B is:
B = M [6.7]
4130143 — bfbi
and C is:
C ___ 2b2 (x _ xmean )+ bl [6.8]
\flibobz — bf
and b0 is:

b = _ 02(46—31/2)

2 [6.9]
106—12y —18

 

163

7—
4

and b; is:

 

 

b0=- y0(’6+23) [6.10]
106—12y -18
andbzis:
— 2—
b2=— 25 32 6 [6.11]
106—12}! —18

In these expressions, y is the skewness, and [3 is the kurtosis. However, there are
constraints upon the skewness and kurtosis for equation [6.5] to produce a real
distribution. If the constraints are not met, equation [6.5] produces complex values which

are not physically acceptable. The constraints on the skewness and kurtosis are6:

0 -< y2 4 32 [6.12]

and,

3
> 391/2+48+6-(y2+4)/2
32—y2

 

.3 [6.13]

The constraints should not be interpreted as meaning that values of skewness and kurtosis
outside of these constraints are not physically possible, simply that equation [6.5] does
not work when the constraints are not met. In order to present these constraints in a more

intuitive manner, Figure 6.33 shows a plot of [3mm as a function of y.

164

Experimenting with MATLAB shows that the values of [3min calculated for Figure 6.33

are somewhat low, and typically about 3% extra must be added to [3min in order to avoid
numerical issues in the calculation, which results in small imaginary parts being present

in the calculated distribution.

However, Figure 6.33 does show that the Pearson Type-IV distribution should be useful
for investigating the ranges of skewness and kurtosis found in the Fabry-Perot films for
this research, although modeling using the exact statistics measured from the distributions
may not be possible. Although RB2K8 has statistics that meet the constraints shown in
Figure 6.33, FB26 shows a kurtosis of less than 3, which cannot be modeled by equation
[6.5]. Also, RB2K9 shows skewness on the order of 0.4, but a kurtosis on the order of
3.2. A skewness on the order of 0.4 means that kurtosis should be at least 3.3 for
equation [6.5] to be used, so the exact statistics extracted from measurements on RB2K9
are not within the acceptable range to be modeled with the distribution in this section.
However, using values close to those extracted from the RB2K9 AFM measurements in

the optical simulations can still provide some insight into this work.

165

 

Min. Value of Kurtosis

 

 

 

l 1

2.5 1 1 1 1 1 1 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Absolute Value of Skewness

 

Figure 6.33 Plot of the constraint on Kurtosis as a function of Skewness for Equation [6.5].

Figures 5.9 and 5.12, which are measurements on diamond window 5, serve as a basis for
the investigation into higher order moments. This film is on the RB2K9 wafer. Figure
6.34 shows a fit of the Pearson Type IV distribution to the measured distribution of
Figure 6.24. The parameters used in this calculation are a mean value of 72nm, a
standard deviation of 22.5nm, a skewness of 0.3172 and a kurtosis value of 3.75. The
value for kurtosis is not exactly the same as the measured value of 3.0308 since the
measured values of skewness and kurtosis do not fit the constraints on equation [6.5].
Notice, though, that the Pearson distribution fits the measurement much better than the

Gaussian distribution in Figure 6.24.

166

 

 

1 Measured Dist. (AFM)
—e— Calc. Pearson Dist. -

 

 

 

 

 

 

 

meters -8

Figure 6.34 Measured distribution from AFM Image 11 compared to calculated Pearson Type-IV

distribution.

Figure 6.35 shows the transmission through the diamond film alone, analogous to
measurement and simulation shown in Figure 5.9. Using the pure Gaussian distribution
in the simulation in Chapter 5, this film was determined to be 1.06pm thick, and have a

surface roughness of 18nm RMS.

167

 

 

 

 

 

 

I I I I I Cl I I
0.9- — simulated _
A 0 measured
0 C .‘
0.85— -
0
0 .
g? 0.8- ‘ . ‘ Q)
5 1
'17 0.75— “
5 e
.5 .. 4-
g 0.7 t 4 ’ -
E, c '
5
I: 0.65‘l . T
1- 0 ' .
0.55 " 4
0 . O
o 1 0
0.55- , ‘ * ° ~
I) _.. ‘ O
1 1.1 o 1

 

 

 

:10 900 1000 1100 1200 1300 1400V'1500 1500
Wavelength (nm)

Figure 6.35 Measured and simulated transmission through a diamond window, simulation includes
surface roughness, skewness, and kurtosis.

The simulation in Figure 6.35 shows uses a skewness of 0.38 and a kurtosis of 3.4. The
skewness is the average of the measured values for RB2K9. The kurtosis is more than
the measured value of 3.17, but it is the closest value which satisfies the constraints on
equation [6.5]. Using these values of skewness and kurtosis in the simulation, the mean

value is fit to be 1.069um. The surface roughness, again, is 18nm RMS.

Figure 6.36 shows a measurement and simulation analogous to Figure 5.12. Here, the

simulation includes the exact values of thickness, surface roughness, skewness and

168

kurtosis given in the preceding paragraph, but the simulation now has gold coatings

included in the calculation.

 

I
— simulated
0 measured

 

0r

 

 

0.06

r
O

 

0.05 "

100%)

0.04

I

0

0.03 -

Transmission (1
I

e
o O
O

0.02

I

0.01 O

I

I, O
0000’ e

 

 

 

1 1 00 1200 1 300 1400 1500
Wavelength (nm)

Figure 6.36 Measurement and simulation of a Fabry-Perot resonator, with surface roughness,
skewness, and kurtosis considered.

The thicknesses of the gold films used in this simulation are 23nm and 27nm as they were
in the simulation for Figure 5.12. The most notable difference between this calculation
and that of Figure 5.12 is that the presence of the higher order moments in the
distribution has shifted the mean value of the film thickness, and the peaks in the
simulation now align better with the measurement than in the Chapter 5 simulation. The

Q values are similar in this simulation to those in Figure 5.12, around 19 the longer

169

wavelength peak and 20 for the shorter wavelength peak, as compared to about 20 for
each peak using a pure Gaussian distribution to model the surface roughness.

For comparison, Figure 6.37 shows both simulations for transmission, that is, using a
pure Gaussian, and using a Pearson Type-IV distribution, with the skewness and kurtosis

as mentioned above.

 

 

 

 

 

 

 

 

0'07 - —°— Pure Gaussian
— Higher Order Moments

0.06 - i -
33 0.05 - 1 EL -
8 i
T
5 0.04 - -
C
.9
U)
.9
E 0.03 ~ -
8
2
I.—

0.02 - _

0.01 ..

1 l I 1 '
1000 1100 1200 1300 1400 1500

Wavelength (nm)

Figure 6.37 Comparison of simulations using a pure Gaussian, and a Pearson Type-IV distribution.

6.8 Conclusions

In this section, AFM was employed to directly measure the surface heights of the
diamond films. From this surface height data, distributions can be calculated. Most of

the measured distributions showed nearly ideal Gaussian behavior, although small

170

amounts of skewness and kurtosis were present in every measurement. Previous
simulations had assumed pure Gaussian distributions in modeling the surface heights of
the diamond film. In this chapter, a more advanced distribution was investigated which
included higher order moments. It was found that the addition of the higher order
moments weigh heavily upon the characteristics of the completed Fabry-Perot resonators,
while affecting the optical transmission results of the uncoated diamond window to a
much lesser extent. When fitting the pure Gaussian to the measured distributions, it is
observed that a slightly smaller or larger mean must be used than is measured, depending
upon the skewness of the distribution. The inclusion of skewness and kurtosis in the
optical model results in the extraction of a different mean than the model employing a
pure Gaussian does. The different mean results in a shift of the wavelength of the
resonator, which can lead to an improved fit between simulation and measurement.
Further AFM investigation of the RB2K9 sample may help improve modeling results, as
the skewness and kurtosis of this sample cannot be modeled with the Pearson Type IV

distribution as employed in this work.

171

References:
l NIST/SEMATECH e-Handbook of Statistical Methods,
http://www.itl.nist.gov/div898/handbookl, 2003.

2 J. F. Kenney and E. S. Keeping, Mathematics of Statistics 3rd Edition, D. Van Nostrand
Company, Princeton, NJ, 1954.

3 M. R. Spiegel, Schaum’s Outline of Theory and Problems of Statistics, McGraw-Hill
Book Company, New York, NY, 1961.

4 J. K. Lindsey, Parametric Statistical Inference, Oxford University Press, Inc., New
York, NY, 1996.

5 See Chapter 3 Ref. 3 (Campbell).

6 Plasun, Richard, Optimization of VLSI Semiconductor Devices, PhD Dissertation,
Vienna University of Technology, 1999.

172

Chapter 7: Conclusions and Future Work

7.1 Conclusions

Many conclusions can be drawn from the present state of this research. The most
challenging conclusion is from Chapter 5, where the modeling of the device shows that
high Q values are not probable with the as-grown surface roughness of the diamond film.
Since diamond is amongst the hardest materials known, polishing the film after the
deposition is not a trivial undertaking and is beyond the scope of this work. However, it
has been shown that polycrystalline diamond films can be polished to under 2nm RMS
surface roughness‘, so this implies that the device presented in this thesis could achieve

excellent Q values with the addition of some post-processing of the diamond film.

A second important conclusion is that the model presented in Chapter 4 is an
improvement in accuracy over existing models in the literature. It was shown that
present models in the literature improperly solve an integral which leads to a model that
does not accurately track the phase of the wave inside the optical thin film. Additionally,
the model in Chapter 4 allows the use of an arbitrary distribution, which is an
improvement over models which build in the use of a Gaussian distribution. Chapter 6
makes use of this fact by using a Pearson Type-IV distribution to model the surface

roughness, instead of a pure Gaussian.

173

In Chapter 3, the fabrication techniques presented lead to a functional Fabry-Perot device,
and large-area, free-standing polycrystalline diamond films. This work is already leading
to new applications, as the fabrication sequences of Chapter 3 were used to create even

larger free-standin g films, used as electron spalation foils in the MSU Cyclotronz.

Chapter 6 shows an interesting result regarding the statistics of the distribution used to
model the surface roughness of the diamond film. It is observed that the inclusion of
skewness and kurtosis in the distribution used to model the surface roughness can

improve the fit between the simulation and the optical measurement of the device.

7.2 Future Work

This work leaves many interesting avenues to be explored, both in the fabrication of the

device and in the simulation of the optical performance of the device.

7.2.1 Future Fabrication Work

Several new technologies have become available at MSU since the beginning of this
work, which could improve the performance and ease of manufacture of the device. Two
systems are of particular interest, one is an electron beam PVD system with 2 sputtering
sources, and the second system is a PECVD system capable of depositing oxides and

nitrides.

174

The PECVD system is of interest because it offers a low temperature deposition method
of coating the wafer with oxide. Also, this means that during the oxide etch, only the
patterned side of the wafer would need to be etched. Additionally, the use of MEMS
techniques or even a simple shadow mask means that the oxide-etch could possibly be

eliminated altogether.

The PVD system is of interest because of its ability to coat the sample with many
different materials, including dielectric materials. This opens up the possibility of using a
material other than gold to form the partially transparent reflective layers on the sample.
If the loss in the gold could be eliminated, the transmission of the device could be
significantly improved. Additionally, it may be possible to use a stack of materials with
alternating indices of refraction to form highly reflecting, but lossless mirrors. Although
the surface roughness of the diamond film would still be a limiting factor, significant

gains in performance may still be possible.

Another interesting avenue for exploration is the use of ultra-nano crystalline diamond
films. These lO-lOOnm grain size films appear to be somewhat smoother than the
~250nm grain sized films used in this research. Additionally, these films appear to
maintain small amounts of as-grown surface roughness independent of film thickness.
By equation [2.5], the use of a thicker film directly increases Q for an ideal resonator.
This appears to hold true for non-ideal resonators, as the two resonators studied in

Chapter 5 follow this trend. The resonator in Figure 5.12 shows Q values almost double

175

those observed in Figure 5.11, and the film in Figure 5.12 is almost twice as thick as that

in Figure 5.11.

Finally, as previously mentioned, polishing the diamond films to 5nm RMS surface
roughness or less should result in excellent Q values according to Figure 5.13. Polishing
the film combined with reflective coatings less absorbing than gold and thicker diamond

films could lead to Q values many times higher than those achieved in this thesis.

 

7.2.2 Future Work with Modeling

Despite the inclusion of the higher order moments in the distribution representing the
surface roughness in Chapter 6, the model developed in this thesis is still relatively

simple. For example, it assumes zero correlation length in the distribution.

Some future avenues to explore with the modeling would be to include a non-zero
correlation length in the simulation. Beckmann does this with a Gaussian distribution,
although none of the references cited in this work for the Pearson Type IV distribution

make any mention of correlation length.

Additionally, near field effects are neglected by this treatment of surface roughness.

More sophisticated models, which incorporate near field effects, may improve the

accuracy of the simulation presented here.

176

Another issue is that the diamond film is considered lossless by this treatment, although it
is known that this is not strictly the case. At a minimum, grain boundaries exist within
the diamond film which may contain sp2 bonds. Other imperfections may well exist
within the individual grains as discussed in Chapter 2. For example, small amounts of
graphite or non-diamond carbon within grains, and the incorporation of small amounts of
feed gasses into the diamond lattice are all possibilities which are not considered with the
present model. The separation of the losses due to non-zero extinction coefficient and

those losses due to scattering are also an important area for future research.

177

 

References:

l J .A. Weima, R. Job, W.R. Fahmer, G.C. Kosaca, N. Muller, and T. Fries, “Surface
Analysis of ultraprecise polished chemical vapor deposited diamond films using

spectroscopic and microscopic techniques”, Journal of Applied Physics, Vol 89, No 4,
Feb 2001

2 D.K. Reinhard, M. Becker, R. A. Booth, T. P. Hoepfner, T. A. Grotjohn, J. Asmussen,
“Fabrication and Properties of Ultra-nano, Nano, and Polycrystalline Diamond
Mdemgranes and Sheets”, AVS 50lh International Symposium, Baltimore, MD, USA, Nov
3r -7 2003

178

APPENDICES

179

Appendix A: MATLAB Programs

Below is the code for the ‘modexampm’ MatLab program. This program simulates
transmission through a slab of diamond with one rough surface, using the MacLeod

matrix method as discussed in Chapter 2.

clear

% enter the nominal thickness of the diamond in meters

dnom=l.0*10“-6;

% let the incident medium be air

nl=l;

% let the final medium also be air

n3=1;

% diamond will be the middle medium, and it's index of refraction

% will need to be calculated with the sellmeier equation on the fly

% set up the weighting scheme

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% roughness of film (RMS) aka standard deviation
sigma=30*10“—9;

dstart=dnom—(4*sigma);
dstop=dnom+(4*sigma);

dpp=10;

incrp=(dstop-dstart)/dpp;

180

nm=0;

for n=lzdpp+l;
d(n)=dstart+((n—1)*incrp);
term1=((d(n)-dnom)“2)/(2*(sigmaAZ));
dt(n)=exp((-1)*terml);
nm=nm+dt(n);

end

dt=dt./nm;

hold off
figure(l)
plot(d,dt,'0')

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% start the loop to calculate transmission

numpoints=400;

lamstart=400*10“-9;
lamstop=1600*10“—9;

incr=(lamstop—lamstart)/numpoints;

for n=l:numpoints+l;
lam(n)=lamstart+(n—l)*incr;
lamu(n)=lam(n)*le6;
trml=(0.3306*(lamu(n))“2)/(((lamu(n))“2)-0.175*0.175);
trm2=(4.3356*(lamu(n))“2)/(((lamu(n))“2)—0.106*0.106)7
n2(n)=sqrt(1+trml+trm2);

% now we need to do another loop...

Tp(n)=0;

for j=l:dpp;

phiair(j)=(2*pi*nl*(d(dpp)—d(j)))/1am(n);

181

 

air=[cos(phiair(j)) (i*sin(phiair(j)))/n1;

i*nl*sin(phiair(j)) coslphiair(j))];

phid(j)=(2*pi*n2(n)*d(j)l/lam(n);
D=lcos(phid(j)) (i*sin(phid(j)))/n2(n);
i*n2(n)*sin(phid(j)) cos(phid(j))];

BC=air*D*[l;n3l:
B=BC(1);

C=BC(2);
t=(2*nl)/((nl*B)+C);
Tplnl=Tplnl+ldt(j)*t);

end

Trlnl=Tp(n)*conj(Tpln));

end

figure(2)

% This will now plot the transmission

plotllam,Tr,'r')
xlabell'Wavelength')

ylabel('Transmission')

title('modexamp.m')

182

Below is the code for the ‘tranmat.m’ MatLab program. This program simulates
transmission through a slab of diamond with one rough surface using the transfer matrix

method.

clear

% enter the nominal thickness of the diamond in meters
dnom=l.0*lO“-6;

% let the incident medium be air
nl=l;

% let the final medium also be air
n3=l;

% diamond will be the middle medium, and it's index of refraction
% will need to be calculated with the sellmeier equation on the fly

% set up the weighting scheme
%%%%%8%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% roughness of film (RMS) aka standard deviation
sigma=lO*lO“-9;

dstart=dnom—(4*sigma);
dstop=dnom+(4*sigmal;

dpp=30:

incrp=(dstop—dstart)/dpp;
nm=0;

for n=lzdpp+l;
d(n)=dstart+((n-l)*incrp);
term1=((d(n)-dnom)“2)/(2*(sigma‘2));
dt(n)=exp((-1)*terml);
nm=nm+dt(n);

end
dt=dt./nm;
hold off

% start the loop to calculate transmission
numpoints=100;

lamstart=1000*lO“-9;

183

 

lamstop=l600*lO“—9;
incr=(lamstop—lamstart)/numpoints;
for n=1:numpoints;
lam(n)=lamstart+(n-l)*incr;
lamu(n)=lam(n)*1e6;
trml=(.3306*(1amu(n))“2)/(((lamu(n))“2)-0.l75*.l75);
trm2=(4.3356*(1amu(n))22)/(((lamu(n))“2)—.106*.106);
n2(n)=sqrt(l+trml+trm2);
rlR=(l—n2(n))/(l+n2(n));
r1L=(n2(n)-l)/(1+n2(n)):
t1R=(2)/(n2(n)+l):
t1L=(2*n2(n))/(n2(n)+1):
W12=(l/th)*[1 -rlL;rlR th*t1L - r1R*rlL];
r2R=((n2(n)-1)/(l+n2(n)));
r2L=((1—n2(n))/(l+n2(n)));
t2R=((2*n2(n))/(n2(n)+l));
t2L=((2*1)/(n2(n)+1));
W23=(l/t2R)*[l -r2L;r2R t2R*t2L — r2R*r2L];
Tmp(n)=0;
Tmp2(n)=0;
for j=lzdpp;
phiair=(2*pi*n1*(d(dppl—dlj)))/lam(n);
U0=[exp(i*phiair) 0:0 exp(-i*phiairll:
phid=(2*pi*n2(n)*d(j)l/lamln):
U1=[exp(i*phid) 0:0 exp(-i*phid)];
S=U0*W12*U1*W23;
Tmpln)=Tmp(n)+(dt(j)*t);
end
Trlnl=abs(Tmp(n)*conj(Tmplnlll;
end
figure(l)
plot(lam,Tr,'“-')
xlabel('Wavelength')

ylabel('Transmission')
title('tranmat.m')

184

Appendix 3: Sample Data

Below is a table summarizing most of the samples created in the process of this research.

Additionally, a series of samples prefixed by “”FB are listed. These samples were

deposited as part of a project to create electron spalation foils. The samples are listed

here because one of the samples is referenced in Chapter 6 in regards to AFM

measurements, while data from other FB samples are used in Chapter 5 as a plot of

deposition temperature as a function of deposition pressure.

 

Name

Deposition
Pressure
(Torr)

Gas Flows

(Hz/COZICI-L)

Depostion
Time
(hours)

Weight
Gain (mg)

Substrate
Diameter
(inches)

Comments

 

Rogdav01

9

200/0/5

20

14.0

3

0.111111 diamond
powder, 53mm
quartz ring

 

Rogdav02

200/0/5

20

8.8

0.1um diamond
powder, 53mm
quartz ring

 

Rogdav03

200/0/5

20

7.2

0. 111m diamond
powder, 55mm
quartz ring

 

Rogdav04

200/0/ 5

40

21

0. 111m diamond
powder, 55mm
quartz ring

 

Rogdav05

200/0/5

20

9.3

0. 111m diamond
powder, 55mm
quartz ring

 

Rogdav06

200/0/5

20

8.2

0.25 pm diamond
powder, 55mm
quartz ring

 

Rogdav07

200/ 8/ 3

40

5.9

0. 111m diamond
powder, 55mm
quartz ring

 

Rogdav08

 

 

 

200/0/5

 

20

 

13

 

 

0.25 pm diamond
powder, 55mm
quartz ring

 

185

 

 

Name

Deposition
Pressure
(Torr)

Gas Flows

(Hz/COZICl-L)

Depostion
Time
(hours)

Weight
Gain (mg)

Substrate
Diameter
(inches)

Comments

 

Rogdav09

7

200/0/5

20

4.2

3

0.25 urn diamond
powder, 55mm
quartz ring, H2
plasma pre-
treatment for 2
hours.

 

Rogdav 10

200/0/5

20

0.25pm diamond
powder, 55mm
quartz ring,
sample broken
when removed
from system

 

Rogdavl l

200/0/5

20

8.4

0.25pm diamond
powder, 55mm
quartz ring

 

Rogdav12

200/0/5

20

0.25 pm diamond
powder, 55mm
quartz ring, bell
jar broken during
deposition

 

Rogdav l 3

200/0/5

20

0.25pm diamond
powder, 55mm
quartz ring

 

Rogdav l 4

200/0/5

20

9.1

0.25pm diamond
powder, 55mm
quartz ring

 

RB2K-2

Diamond on
oxide, 59mm
quartz ring

 

RB2K-3

No deposition

Fabricated oxide
windows, 59mm
quartz ring

 

RB2K-4

4.1

59mm quartz
ring

 

RB2K-5

10

200/0/5

15

3.2

59mm quartz
ring

 

RB2K-6

35

200/0/5

10.9

59mm quartz
ring

 

RB2K-7

10

200/0l5

15

3.9

59mm quartz
ring

 

RBZK-S

35

200/8/3

5.5

8.7

59mm quartz
ring

 

 

RBZK-9

 

35

 

200/ 8/ 3

 

5.5

 

6.3

 

 

59mm quartz
ring

 

186

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Name Deposition Gas Flows Depostion Weight Substrate Comments
Pressure (Hz/COlel-L) Time Gain (mg) Diameter
(Torr) (hours) (inches)
RB2K-10 35 200/8/3 3 3.8 2 59mm quartz
ring
RB2K-11 35 200/0/5 3 10.9 2 59mm quartz
ring
Table 3.1 List of samples created for this research.
Name Deposition Gas Flows Depostion Weight Substrate Comments
Pressure (Hz/COZ/CH4) Time Gain (mg) Diameter
(Torr) (hours) (inches)
FB-2 29 200/8/3 6 10.1 3 59mm quartz ring
FB-3 30 200/8/3 6 8.4 2 59mm quartz ring
FB-4 30 200/8/3 3 4.7 3 59mm quartz ring
FB-S 33 200/8/3 6 3 59mm quartz ring
FB-6 34 200/8/3 4 8.5 3 59mm quartz ring
FB-7 35 200/8/3 4.5 9.0 3 59mm quartz ring
FB-8 35 200/8/3 3 2 59mm quartz ring
FB-9 35 200/8/3 3 2 59mm quartz ring
FB-lO 33 200/8/3 4 2 59mm quartz ring
FB-ll 33 200/8/3 4 2 59mm quartz ring
FB- 12 32 200/8/3 2 59mm quartz ring
FB-13 32 200/8/3 3.5 3 59mm quartz ring
FB-l4 33 200/8/3 4 59mm quartz ring
FB-15 33 200/8/3 4 59mm quartz ring
FB-16 33 200/8/3 6 59mm quartz ring
FB-17 33 200/8/3 6 59mm quartz ring
FB-18 33 200/8/3 5 59mm quartz ring
FB-l9 33 200/8/3 5 59mm quartz ring
FB-20 30 200/8/3 5 59mm quartz ring
FB-21 30 200/8/3 5.5 59mm quartz ring
FB-22 30 200/8/3 5 59mm quartz ring
FB-23 33 200/8/3 7 59mm quartz ring
FB-24 33 200/8/3 7 59mm quartz ring
FB-25 29 200/8/3 7 59mm quartz ring
FB-26 20 200/8/3 12 2 59mm quartz ring

 

 

 

 

 

 

 

187

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Name Deposition Gas Flows Depostion Weight Substrate Comments
Pressure (Hz/COleIL) Time Gain (mg) Diameter
(Torr) (hours) (inches)
FB-27 20 200/8/3 l6 3 59mm quartz ring
FB-28 20 200/8/3 26 3 59mm quartz ring
FB-29 6 200/8/3 48 3 59mm quartz ring
Table 3.2 List of F3 samples.

188

 

Appendix C: Additional Measurements

This appendix contains the optical transmission measurements made on the various

samples for this research that were not included in Chapter 5.

 

 

0.85 - .
0.8 - .

:9

§ 0.75 - -

Z

O

5 0.7 . .

.52

E

a)

5

,_ 0.55 - 4
0.6' .
0.55 - .

 

 

 

600 700 800 900 1000 1100 1200 1300 1400 1500 1600
Wavelength (nm)

Figure C.l Diamond window on the R32K9 wafer measured with Perkin-Elmer UV-Vis system.
0.66pm thickness.

189

 

 

0.95

I
I

0.9

I
I

0.85 -

100%)

0.8 - a

0.75

Transmission (1

I
l

0.7

I
I

0.65

0.6

q

 

 

L I I

I I I L J I
700 .0 9CD 1111'] 1100 1200 131]] 141]] 1500 1“
Wavelength (nm)

 

Figure C.2 Diamond window on R32K9 measured with Perkin-Elmer UV-Vis system. 0.695um
thickness.

190

 

 

 

 

I I I I 3 0
’I
0.8 "' 1' 0 I
’33 t. 1‘
0.75 - '. 1‘ .1
II
’1‘ ’0
A ‘0 1.
°\0 0.7 '- w I. u
o
o '0
,—
II 1' t. \I
,—
V 0.65 " 1| 4. 1. "
C
.9 ..
'D
.8 1'
g 0.6 " ‘. '. '. 1‘ ‘l
g I. 1) '0 1‘
l— 'D
0.55 " 0 \I I
1. 1‘ '0
I. “
0 ,‘ w
o 5 \‘
< I. \‘ "
\I ‘ ¢;6‘
1 1 1 1 1 .
1100 1200 1300 1400 1500 1600
Wavelength (nm)

Figure C.3 Diamond window on R32K8 measured with Bausch & Lomb system. Approximate
thickness 1.55m

191

 

 

 

 

l l I T I 1'
‘1‘
’0 1‘
0.95 " 0
’0 1‘ I.
I. 0 'D D
0.9 b d
g '33 '0 0 'D
8 1‘
v- 4.
'u 0.85 I. “ ‘D “ ‘. _J
C
'3 4' v o
I \ I
g 0.8 _ o D _J
m 4
g a a o D
r- » .
0.75 P- 1. ' .
I. ’D
\
'D ‘ 1!
0 7 “ D D
. " 1‘ , '
< " \‘ I‘ c ‘
6" s‘ - ' I
v‘
Q . L L l l L
1 100 1200 1300 1400 1500 1600
Wavelength (nm)

Figure C.4 Diamond window on RBZKS measured with Bausch & Lomb system. Approximate

thickness 1.62pm.

192

 

 

0.09

l

0.08

0.07

1 00%)

0.06 -

0.05 -

Transmission (1

0.04 -

0.03 -

0.02

 

 

 

 

1050 1100 1150 1200 1250 1300 1350 1400 1450 1500 1550
Wavelength (nm)

Figure C.5 FP Cavity on RB2K8 as measured with Perkin-Elmer UV-V'B system. Diamond film is
1.59m thick, with 12nm and 20nm gold coatings.

193

 

 

I

0.05

0.045 -

0.04 -

1 00%)

0.035 -

Transmission (1

0.03 1

0.025 -

 

0.02 -

 

 

l l

1 100 1200 1300 1400 1500 1600
Wavelength (nm)

 

Figure C.6 FP Cavity on RBZK8 as measured with Perkin-Elmer UV-Vis system. Diamond film is
1.55m thick, with 12nm and 25nm gold coatings.

194

 

 

 

 

 

, :1“
0.9 - ‘ -
1‘ 1‘
r‘ 1‘
0.85 - "‘ " «
l. “
1‘
0.8 _ ¢1‘ D 1
3 ID 1‘ 1‘
08 1‘
41
'fi 0-75 - ., n o -
C 1‘ a 'D
s 1‘
'35 0.7 - .. -
.9 a
g 1‘ 'D ” 1‘
g 065 I. 'D 1‘ D 1‘ d
a
06 l. 1‘ 1‘ b “ .
'D 1‘
1‘ ’0
1‘
0.55. D 1‘ ‘
1‘ ‘ ¢;;‘
1100 1200 1300 1400 1500 1600
Wavelength (nm)

Figure C.7 Diamond window on RB2K8 measured with Bausch & Lomb system. Approximate
thickness 1.55pm.

195

 

 

 

 

0.075 - r ' ' ' T q
1‘
0.07 - .
to

0.065< “ .
:\°‘ 1‘ 'D 1‘
g 0.06 - , ‘ -
J-I . 0 I. 1
g 0.055 - a .
.‘é’ .. w w
5 0.05 - , .1 -
E . ‘
l— O

0.045 " 1‘ -

1‘
" 1‘
0.04 - c ’ .
0.035 - “ c -
1100 1200 1300 1400 1500 1600
Wavelength (nm)

Figure C.8 FP Cavity on RBZKS measured with Bausch 8: Lomb system. Approximate diamond
thickness 1.75mm. Gold coatings approximately 12nm and 25nm thick.

196

 

 

-

d
.
q

0.26 - r. ' '

 

 

 

a -
(o
0.24 " c 1‘ ‘
0.22 .
A '.
5: 0 2< " " w n .
S2
v|_l
V 0.18 - .
C
.9
z; ..
.E O 16 ,_ I. ‘. 1‘ "‘
U) 1‘
C
(U
l: 0.14 - -
U ‘ '0
012 ° .1
. 5. ’
0
0.1 '- 1‘ ..
g "
l l l 1 1 ¢ - Q
U
1100 1200 1300 1400 1500 1600
Wavelength (nm)

Figure C.9 FP Cavity on RBZKS measured with Bausch & Lomb system. Approximate diamond
thickness 1.57 mm. Gold coatings approximately Onm and 15nm thick.

197

 

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